Do NOT go in “with an offer in hand” and demand a counteroffer. This is blackmail, pure and simple, and can only hurt you in the long term… And in the short term, you will get crap piled on you like you would not believe.
My suggestion would be to have a talk with your boss in a conversation based around the theme that you are “taking stock”, about both your personal life (i.e. your new family addition, the mortgage, possibly more children to come) and your present and future career with the firm.
This is likely the pitch that will get the best reception from management. It is always better for a company to retain valuable individuals than to replace them, and it sounds like you are in this boat. Taking stock of your current and future goals and opportunities is a normal part of career planning and will not be viewed as disloyal, while at the same time, obvious conditions for dissatisfaction (reduced pay coupled with increased financial straits) will get seen as a potential way to lose you that the firm will want to avoid, rather than an overt threat of abandonment on your part.
In general, you should have this kind of talk at least once a year, if not semi-annually, with your supervisor. From the other direction, as someone new to management, I have been instructed to make sure to initiate these discussions with the people under me, to make sure we always have a good feel for who’s potentially unhappy (the worst thing for a manager is to have someone they rely upon disappear unexpectedly).
If the company is really hard up for cash, they will at least try to give you other benefits, perhaps more time off, giving you an assistant, etc.
If they get all riled up and essentially tell you that you can go walking if you want to, well, leave. They’re either idiots who can’t run a company (in which case you don’t want to stay there anyway), or they didn’t really think of you as someone with a future at the firm (in which case you don’t want to stay there anyway).
But then you get to go job hunting (a) with a clear conscience, and (b) doing it from your desk while getting paid, because they certainly can’t fire you just for discussing your career.