I think thevissue is time. He just has his lunch. Stopping to change and leaving time to change back will cost him a big chunk of his time.
The email from the company specifying dress code is an odd one, for such an informal company, so I suspect you’re right to think it’s targeted at youth who don’t know better.
I always think ‘dress for a client meeting’ in these situations, where the clients know we dress casually, but expect us to be professional. I work in the creative industry, where the creatives tend to be super casual, but the account managers dress better because they meet clients all day. Not super smart suits, but certainly clean, pressed collared shirts, nice jeans, leather brogues and a tailored jacket.
If this is how you normally dress (and it sounds like you do), plus you’re at a senior-ish level who knows the norms for your trade, then I think that sounds fine.
You could always drop in a casual comment with a wry smile along the lines of 'I would have worn a suit, but my colleagues would have smelled a rat".
It also assumes he has a car. If he takes a bus or train downtown, carrying a suit, hanging it in his cube, changing into it to walk or grab a cab to the interview - it isn’t any more discrete than just wearing it to the office.
Even at the cost of 15 min of an hour long interview?
BTW, this was for a job as an editor of an industry publication, and while prepping for the interview, I remembered that the editor-in-chief had once won a Pulitzer Prize. So I went to the library and found a book that contained previous Pulitzer Prize-winning articles and read his. So when he mentioned as I was leaving about how horrible the traffic was on a particular highway, I casually mentioned that he’d said something about that in his prize-winning editorial. I didn’t get the job, but I was proud of myself for making me memorable.
Right (to Manda JO). I work at a casual tech company, and it actually tends to be a strike against someone if they show up for an interview in a suit. It suggests they don’t know the industry, and they haven’t bothered to do any research about our culture. More subtly, it gives the impression they need to compensate for some deficiency in their skills or experience. When a candidate shows up in a suit, my thought pattern is, a strong candidate who has worked in tech around here wouldn’t be caught dead in a suit. So why does this guy think he needs one?
Too late for the OP, but your plan sounded perfect. Good luck.
Wear it to work. If questions are asked, come up with a plausible answer. Even, perhaps, the truth.
The advice I’ve always heard is that you should dress one step nicer for an interview than you would for the job itself.
I’d just show up. They know you’re coming over your lunch break so they should expect you to be in your usual working clothes. Now I’d make sure that you were wearing your newest normal work shirt and pants or possibly buy some new ones so they’re in great shape. Do you really want to work some where that will schedule a lunch interview and expect you to specifically dress up beyond normal work appropriate.
I wore a suit to a job interview almost three years ago but that job would require me to wear a suit most days. The rest have all been lunch interviews. I wore a jacket on a job interview about 5 or 6 years ago and I actually got made fun of by my interviewer (still got the job though).
The tech industry may be an exception. I heard you are supposed to wear jeans to a Microsoft interview. I never bothered to verify that. Perhaps that started a trend.
I’m late to the party, but I agree with everyone who agreed with your plan. A quick, lighthearted comment to remind them that you’re on your lunch hour is fine.
However…you’re interviewing at a start-up in the hopes of a better work life balance and fewer hours? I’ve never heard of it working that way, personally. 
Not a startup - it’s an established SaaS company in the niche it plays in.
Anyway, I threw on a sportcoat in the parking lot over the dress shirt / jeans / loafers combo, and I was the sharpest person I saw there. Bowling shirts and shorts seem to be de rigueur.
Also, I appear to have knocked it out of the park.
Congrats!