My sister: “I never eat anything that used to have feelings, and I would never wear leather or fur!”
Me: “Yes, dear, but I am not a nice person, and I just don’t give a fuck.”
My sister [laughing]: “OK, I can never get mad at you.”
Again I wish I could rate comments in this forum. 
You and your kid’s baby mama (you didn’t say spouse) have what appears to be terrible communication.
You assume that she agrees with you that he has “certain behavior problems”. Maybe she doesn’t. So discuss that first. If you both agree that he has “certain behavior problems”, then the issue next, is how to address them. One solution might be taking him to see someone. But you and baby mama, never even got to that discussion because she shut you down.
Nope!
Ha! 
Yeah, I admit, my first thought (which I squashed to try to be ‘helpful’) to the ‘you think our son is broken?!’ was:
- “What? You don’t want to get our son help? Don’t you care about our son?!”*
You need a meta-discussion: a discussion about what form of debate is good for a relationship. First you need to establish that this is an equal-sided relationship with trust - do you trust your spouse that they don’t want to belittle you, put you down or play manipulative games in order to win intentionally? (A lot of people learn bad habits from their own families and slip into them unconsciously, but do you trust your spouse when they say "I don’t intend to hurt you with my words?)
Now ask your spouse the same about you - does your spouse believe that you won’t manipulate, belittle etc. them intentionally?
If either one can’t honestly answer “Yes” to this, then your relationship problem and trust problem needs work first (I’d suggest an external counselor; a mediator or counselor might also be a good idea to have present for this first meta-talk. This needs to be somebody completely neutral, meaning ideally nobody related or friends, so there’s no accusation of taking sides; and professionally trained to stay neutral and not get drawn into personal accusations if the talk goes bad).
If however you both have trust and your relationship is not intended to put one part down - then you need to give the “I” feedback: “When you say … I feel as if …”. Speak about your feelings, but don’t act on them. This means that you need to take a few hours or days to cool off from the last argument and explore your feelings in the meantime.
Basically, a person using that shitty an arguing style is either playing to win intentionally, in which case you need to re-evaluate your relationship; or learned bad habits, so explaining why this type of arguing is bad would help.
In the two examples you gave, further information and feelings would help. Like doing feedback questions: You feel irresponsible for using money for non-essentials? (Maybe they have bad experiences in the family or with friends spending irresponsibly). But look, we’ve drawn up a budget plan, income minus expenses, mortgage payoff is this sum, savings account is this amount, and 100$ each month are left over, so if we save for two months, we can buy a small new TV and still meet our obligations.
Why do you believe that a child / person can either be broken or perfectly allright? Do you feel that taking a child for professional help is an accusation against your skills as parent? I can understand that fear, but here’s why this is not true…
Note: this amateur psychological trouble-shooting to find the real cause (feelings or misinformation or both) only works if you are honest and not manipulative, either. After all, there’s a possibility you are wrong, or there’s a third option. The goal is not to be proven right - for either of you! - but to find the best solution for your situation. Both of you need to remember that.
If you both agree that you need to work on your communication style, you could when talking with the counselor also agree on code words or signals (maybe clue cards) to defuse the situation. Saying “This makes me feel angry, I need a time-out” or “You are shouting, I don’t want to shout back, let’s calm down” or “I feel manipulated, let’s take a step back and look at the meta” or whatever. Again, not as a tool but as an aid.
Unless the person is so stupid and lazy-thinking to look at everything in black-and-white. But then you would have mentioned it, I guess.
“Looking after the wellbeing of our child is the responsible thing to do. Don’t you wish to do the responsible thing?”
I, too, know someone that takes the moral high ground with everything relating to money and self-discipline. I’m frugal, but this person can’t even buy a piece of gum without analyzing the pros and cons. When I got my son potty trained, instead of being happy for me, they went on and on about the savings in money from not buying diapers. I’ve got the a/c on at 79? I hear how they save money by keeping it as 80. I get a pedicure? I hear how they is “frivilous and ridicoulous”… I get my hair highlighted, they tell me how their spouse colors it for them and sings “I’m saving $50” while they do it. I order a pop at lunch, they order the water and proceed to tell me how much $$ they’ve saved from choosing water over pop. Every. single. thing. related to money and their superiority of handling it. They claim everyone else spends too much, doesn’t save enough, doesn’t have enough self-control, etc etc…
I save what money I can, I don’t make a lot, but it’s RIDICULOUS how this person is always morally superior when it comes to money and being self-disciplined. I leave there thinking I’m a glotton and a failure.
“I’m going to do my part by the future by sterilizing you in your sleep, so your hellspawn won’t chow down the planet’s resources for eternity.”
Ah, see, that’s because you’re a nicer person than I am!
OP, in addition to taking Lady’s advice (and not mine ), go read constanze’s post again. You both need to work on your communication styles and skills.
Actually, I don’t.
Care to clarify?
A glutton and a failure with well-groomed feet and nice-looking hair.
I think you need to go on a counteroffensive here, not just to put this person in her place, but also for your own self-esteem.
“I wish I could go ungroomed too, but it’s considered really unprofessional to have disgusting gnarly feet and obviously amateur hair at my job. And it costs money to replace shirts with ugly yellow sweat stains like the ones you have.”
Okay, don’t say that exactly. 
Who is this person and why do you even hang out with her?
I don’t think the OP’s examples have anything to do with “taking the moral high ground”. He’s just dealing with a spouse who either likes to argue (by presenting contrary positions or ideas, not necessarily based on anything ethical) or has no confidence in his decision-making. Such behavior is pretty hard, if not impossible, to change.
But in a debate team (you mean those in US High Schools, yes?) people are taught to win, not to discuss with facts. In fact, using a bad debating style wins better.
What you both need to learn is how to communicate, which is completely different from winning a debate.
Actually, I don’t think debates on a non-personal level work at convincing, either. If you start out wanting to score points or to convince the other person that you are right, then the automatic human reaction is to dig down and defend your own position, without stopping to examine valid points that might be made, or real flaws in one own’s logic. So a friendly discussion starting out with “What facts and ethics and feelings do we have about topic X? Lets both compare” in a friendly atmosphere with somebody you trust and who isn’t insane - that can achieve the goal of considering other viewpoints and learning new facts or different angles.
That in turn might lead to a reconsideration.
But angry, confrontational TV-style debates are only good for preaching to the choir, scoring points with low tactis and yelling a lot.
I’d say that depends on the person’s willigness, trust and introspection. Most of us learn debating behaviour from our parents and friends. If somebody never learns to look at their feelings and analyze and express them (I feel angry when you say … I feel belittled when you call me… I feel that you imply I’m an idiot for not …) - then they can only yell in anger “you’re an idiot, too”.
If somebody has always seen relationships being about power and dominance, and every talk about an opportunity to put somebody down - then they don’t even realize that things can be different.
However, if they have self-reflection and introspection, they are able to understand a meta-talk about how different and better patterns of behaviour would look like and get to start working on them with friendly encouragement.
The key there is the trust that the other partner won’t use it as weapon - as some non-nice people do “There’s something wrong with you, so I will give you feedback and disregard anything you said”. Always use a neutral voice, not be angry yourself, when giving feedback, call for a timeout and do resume the discussion later and try to take the points seriously.
And if you try to find out the feelings involved in a decision, and your spouse denies them, accept it. Don’t follow Freud “the more you’re denying, the more it’s true because it’s just buried deeper”, accept what they are telling you how they feel and see what the best course therefore is.
Basically, I do believe that people willing to change can get rid of bad habits; but it is hard work.