I’ve been where you are. Except my SS mum also wanted to inspect my home before my SS was able to stay.
I acquiesced to the request in the hope it would make the relationship easier long term and to ease what was a very difficult relationship at the time. Even though the same good grace was not given in reverse! She had already moved their child into a new home with her partner without consulting his father at all and I had been with his father longer than she had been with her partner.
I personally feel, looking back on it now, it was easier than continuing the aggressive arguments at the time and allowed my now DH to continue his relationship with his son unhindered. Therefore I am not sorry I did it overall and I had nothing to hide so it didn’t really matter and I would do it again, if asked, to grease the wheels.
That being said, in our situation, over 10 years later, I now realise it’s a matter of control. She likes to have overall control and can be quite manipulative with my SS if necessary. He’s such a sweet boy, he just wants to please so it can be quite sad and now as she has split from her DH and moved back in with her mother (who is super manipulative of her and SS, which is part of the reason for the breakdown of the marriage) things have become awkward at times as her DH used to be quite good at mediating how he would feel in my DHs situation which was a help.
Her ex-DH has called us to say she is seeing someone new already. She then called us to discuss this and I was clear that as his mother we are confident she would keep SS safe when with her like we would expect her to take that view with us, so we had no interest in the situation whether he was a new partner or not.
We have always tried to take the high road and broad view, she’s been to our new home and I have welcomed her (although she didn’t look round this time), and we’ve had her other son for an afternoon, etc. because I am conscious that as SS becomes older and moves into adulthood he will see things for what they were and I want him to see that we always tried our best to do the right thing to make things as workable as possible for his benefit. Sometimes we, as the adults, have to be responsible enough to look at the long term.