Should a person with mild dementia in AL, go home for a week to “see how it goes”?

Where was this PCP when she is in the AL? Once she is there, her PCP should become her former PCP.

My brother was diagnosed, after a car accident, with probably early Lewy's Dementia. He was diagnosed by his symptoms, not by any testing or exams, or MRIs. He at that time lived in a very small home with very close and helpful neighbors. His worst problem was his balance. We went to rehab, and then together we made decisions based on recommendation. At the end of his rehab he was so good that they said they believed he could return home with support. However, while I could be his POA and his Trustee of Trust, taking over all record keeping and paying bills, I lived at the other end of the state, and was, like him "of an age". My bro wasn't at all computer savvy and could not order groceries in that way. Only one store in his Palm Springs area could take orders by phone and deliver. I could deliver staples through Amazon. In the end we chose Assisted Living, where he already had an ex-partner/close friend. He got two rooms and we moved him, keeping his small home. After about two months he said he thought he would like to return home. I told him this had to be his decision and that I would assist all I was able, but I asked him to keep the ALF two rooms he had out of fear of his changing his mind, and our losing those rooms. He went home to supervise a new roof, but realized during that time that without a car he was dependent on the "kindness of strangers" more than he would like to be. He also realized how unstable on feet he was becoming because of balance issues. He decided ultimately not to make this move and to sell this home, which is what we did together with the help of a dedicated realtor. It was the right decision. I was very relieved that he could make it on his own, because that is the best way.
If you do this/attempt this, I think it is crucial that you not "enable this" and that you pretend that YOU live not only 40 minutes away, but rather that you live a state away. IF you enable it I believe you both will go down under it. Make it clear that your Mom should try this out, and that you will check on her, perhaps become POA and handle her bills, etc. But that you will not be seeing her more than once a week on a weekend, and you will not be chief cook and bottle washer.
I also think it is crucial that your Mom is of somewhat solid state of mind, rational, as my bro was. It is of concern to me that the police had a feeling she was not, clearly, and don't know the details.
I wish you good luck. Don't give up her room if she has a good one. And hope that she comes to the realization on her own that this isn't the right move for her. And again, if her mind is not sound enough this move/this experiment should be out of the question.