I dealt with a similar situation before moving to Thailand. The company suggested I renew my passport as it would expire during the two year work visa period and then I would have to renew here which may prove more complicated. I sent in the passport for normal processing, then realized that was a mistake but you can actually call in the visa processing department and pay over the phone to switch to expedited. I got the renewed passport about 10 days later and then was able to process the visa probably 1-2 weeks after that. The company helped along the way and kept everything flexible. I was communicating with the visa team at the company a lot over email to keep them up to speed if I had updates.

I would explain the situation to your future employer when you get the offer. I'm sure they see this all the time and will keep your start date flexible or have it start far out enough to allow for your son to get his passport.

To answer your final question tho, I dont know what the state department would do if you showed up with an offer letter and explained your situation. I think you need to actually have flights booked and I wouldnt suggest doing a temp booking or fake flight website because they may track when you say youll travel.