OP, can I suggest that you talk this situation through with your daughter when you meet up with her? Tell her that while you would love to have your grandchild for one or two days a week (or whatever suits), you are not going to be emptying your spare room. Her child needs to learn to fit in with normal family life, and as time goes by, be taught that there are things he can touch, and things he can’t, and that putting him in a play room won’t teach him those things. Point out that you need that room for other things, which is why you suggested babysitting at her home, but if they're not happy to accept your offer as is, then perhaps she'd better start looking around for a proper nursery.
If she says, but DH is worried that the baby won't be safe in the sitting room with you, because of the cats or whatever, and you really need to make the effort if you want to look after the little one, then tell her that you'll consider clearing the room, once you've got into the routine of having him, as she may change her mind, and decide she can't bear to leave him and go back to work, or the baby may not be happy being with you, or you might find having him too much, so far better to have a trial run, and then once you've established it's going to work, perhaps she can give you a hand to clear the room. Tell her you've offered to care for your grandchild because you love him, and want to spend time with him, but you're not going to rearrange your home, until you see how things go. As things stand, I think it highly likely you'll go to all the effort of clearing the room, only for her DH, to come up with some other excuse why you can't look after the baby.
Depending on your relationship with her, I would also tell her that you are not going to be dictated to by her husband, about what you do when you care for their son, and ask her why she has allowed him to ban you from the house, until you agree to his BLACKMAIL!! Ask her if he controls her too. Does he not like her spending time with other Mum's, maybe puts obstacles in the way, but of course, only because he 'knows' that they talk about her behind her back, or they're not good for her, or some other ridiculous excuse that he comes up with. Ask her if he's doing his fair share of child care, and whether he has a night out, and if so, whether she gets one too. Obviously only you know if you can be blunt with her, or whether you need to tread carefully for fear of her stomping off in a paddy, because you've criticised her husband, but you do need to let her know that you’re not blind to his controlling habits, and that if she ever needs to get away, you will ALWAYS find room for her and the baby.
