You have every right to be concerned and glad you found us here. Several of us have been through this type of situation including me.
My Mom and Dad was very much as your FIL is now. Because of their inabilities and behavior is was necessary to put her (with Alzheimer's) and Dad (with Vascular Dementia) in AL. They both wanted to "go back home". In my years of dementia experience I have never met anyone with dementia that didn't want to "go back home"!!
I also understand the difficulties of knowing what to do regarding our loved ones with dementia who do want to go back home. We want to make them happy. If they just see home maybe they will be more content... perhaps. We don't want to take home away from them. But the question is... are we satisfying our desire to keep the concept of "home" intact or are we don't what is best for the.
First my experience with Mom and Dad. Once Dad was in AL we never took him back to the house. He adjusted well to AL. He thought it was temporary and for Mom. He never speaking of "home" but he was content where he was. Mom on the other hand, my sisters decided it was imperative to take her back to see the house. The first time she went back sent her into a depression when she left once again. It was actually worse than the first time she left. The second time she went back to the house it was as if she was not sure where she was. As we were leaving she ask to go home. So where was "home" for her at that point? The third time we went to her home town and had lunch with some of her friends, she was scared, stayed by my side all afternoon, and had no idea that there was even a "home" in that town. Non of the trips ended well.... and as soon as we returned to AL she was talking about going HOME!
You have to put yourself in your FIL's frame of mind to understand. The anxiety of dealing with a house, bills, and day to day living in that house was too much for him and probably part of what lead to his behavioral problems. You have taken the steps need to get him help and find a place that he is relatively content. If he thinks that is temporary... that is ok. Let it be temporary as long as need be. It will eventually become his home in his mind.
If you do take him back home he will respond in two ways and neither of them is good for him. He will either respond as he did the first time he left with anxiety and depression or he will not know why you took him to that house and be anxious and depressed because he is aware that you expect him to remember something he can't. Why put him through that anxiety?
Then we have to understand what "home" is to someone with dementia. There are moments they remember their recent past "home"... the house you are thinking of taking him back to. Rather than being fond memories and a nice place to visit it brings on anxiety. It is something that is lost. If found they want to hang on to that again. So why give it and take it away repeatedly. At other times they may be thinking of "home" as their child hood home. Their long past memories come back, blend with the moment, and they wish for that comfort and security of their childhood home. Which bring on the third "home". This is the "home" which only resides in the safety and security that desire where they are free of the confusion and anxiety that their dementia causes. This is not a place but a state of being that you can not give them. The last "home" is their heavenly home. I have been able to asses Mom and Dad's desire to go "home" to each of these places. So so not assume that going "home" means going back to the house they came from. Even if it is the brick and mortar house they came from... why take them back only to take them out again enhancing the anxiety they fell.
From experience, my recommendation, ff your FIL is content in his "temporary" home.... leave him be. Don't add anxiety by taking him to a place he may or may not remember but will surly add to his angst.
Your SIL is probably sad that Dad is no longer able to stay at home. She is probably feeling some guilt because Dad had to leave his home. She is missing her home and Dad being there. I get it because I felt those same feelings. But we have to deal with our own emotional turmoil without taking Dad with us
Perhaps talking to the SIL about her feelings will help. Armed with information about FIL's state of mind you might be able to convince her that it would not be good for Dad... if she knows you are concerned about her emotional well being