Apologies if this has been already been answered, but what kind of financial arrangement does BIL have with FIL when he's living with him?
You can't really say no - and you haven't - but YADNBU to expect there to be some level of organisation and acknowledgment as to the level of disruption this will cause.
At the very least you need:
A fixed date of arrival
A fixed date of departure
An agreed contribution to his keep, if he can at all afford it (And unless FIL is paying ALL his bills when he lives with him, he's had to at least have been buying himself food, so that shouldn't be an issue)
A definite agreement that he keeps to your household routine, and that you will not be doing his laundry etc for him!
On the subject of sleeping arrangements, however, he is a working adult, and the sleeping in the lounge thing will cause more issues for both him and your family then it is worth to make a point, so I wouldn't do that. Either have him share the room with your son, or, yes, move one of the kids.
I would not, though, rearrange entire rooms worth of furniture. One drawer and a bit of wardrobe space is fine and your child still needs access to their stuff when they want/need it.
If he insists on bringing his bed, that's fine, too (solves the problem of finding him something to sleep on!) but if he's taking over your child's room, then he sleeps in that bed and the spare (his) gets put up for your child. You don't want to be moving two beds for his convenience when one will do.
Oh, and he (and your DH) move all furniture, not you.
At the end of the day, he's family, and that means you're stuck with him, however piss-poorly the whole thing has been handled (and it has been - I'd be screaming!) but you're doing him a HUGE favour here.
Send your SIL, FIL and BIL a list of the terms now. Tell your DH that this is how it's going to be and inform him that if he doesn't have the balls to confront the real issues here, then he doesn't get to sulk because you have.
If he's (DH) really prepared to drag your financial worries into this to try to change the argument, well turn it right back at him. If things are so tight that he can use it as an excuse that way, then they're too tight for your children not to suffer for keeping BIL as seems to be expected and that's not happening. End of. Sorry.
Good luck!