I had to rehome my mums dog when she went into hospital and then care home. That was really stressful (i work long hours and have cats so taking him on wasnt an option) but i found him a lovely home with a wonderful lady - he has the life of riley and looks younger and more fit than ever. (I think mum used to feed him inappropriately) i had a dog walker take him out every day but he gets much more exercise now.
I actually got her a cuddly golden spaniel toy from that big toy warehouse. It looked quite lifelike. She loved it for about a year and carried it everywhere with her, looking after it. I even bought spares in case, heaven forbid, it went walkabout (which does happen in care homes!!) as she would have been very upset. Then she lost interest in it. But it gave her a lot of pleasure in that period - i guess it depends where on their dementia journey they are.