You are definitely sensible, Bright, cannot imagine many people remaining as clear-headed as you sound just now.
You are right, of course, about the stranger off the internet thing. Although I was saying that too, I agree it would look pretty bad. I think we've been suggesting it just because we are all so concerned that you should have some support for yourself and an independent witness there, particularly if you get emotional (under the same circumstances, I reckon I'd be totally in pieces) and because we all wish we could do something. But that doesn't make it sensible of course.
It might be worth speaking to one of the homelessness charities, Shelter or someone? There must be advocacy or similar support available from somewhere. Hopefully someone who knows more may make suggestions soon.
But in the meantime, you will at the very least be seeing your son today and you have a solicitor appointment tomorrow. So things are moving in the right direction.
As Pps said, please try to take care of yourself too, as best you can. It can be harder to reach out for support from RL friends, but, well, they're soon going to know you have a new baby, so it must be worth asking for help now. Take care Bright, and good luck for today.