K8eee, I know where you're coming from, and I don't think you're asking too much.
My DD is almost 4 weeks, and I have 3 DSC (9, 5, nearly 4) who spend every other week with us. This is most definitely their home as much as their mum's is...but I asked DH to arrange with their mum for them to stay with her for the first couple of days we were out of hospital, just to give me a chance to find my feet (in the end it wasn't necessary as it was a week they were at their mum's anyway). Most of the time DH & his ex stick rigidly to their custody arrangements, but changes of a couple of days here and there are sometimes made, and for much less reason than a new baby.
That said, we made sure that my DSC were the first visitors to the hospital, the only people we allowed to come on the day DD was born (DH went to get them from their mum's). Is there something like this you could do to show your DSS he is special? (Easier for us as we don't have the distances you do).
The first week my DSC were here after DD's birth it was half-term, and there again, they only slept here, spending the days at their mum's - we'd never usually do that, but DH runs his own business & couldn't promise to be around all week, and I felt I was well within my rights to refuse to look after a newborn plus 3 under-10s.
In all cases, though, I really had to explain, explain and explain again to DH; he just seemed to think that we'd sail through the whole thing, and he was also afraid I was turning against his DCs now that I was about to have my own.
I should point out that for me none of this would be necessary if we had any help with my DSC, but despite my ILs and all DH's close friends (inc DSC's godparents, etc) living locally, we don't. If we could have counted on people to take a child or two for a Saturday afternoon, feed them, take them to tennis lessons, etc etc, the plan would have been for them to stay with us.
All is now back to normal, and although I'm not looking forward to the Easter hols, there's no way I'd ask for the DSC not to come, as circumstances don't justify it any more.
Good luck!