Take the money side out of the equation for the moment. If you are unhappy with the care he is receiving at that care home then yes move him. He probably will deteriorate from a move as that is very common for people with dementia, but he might not, or it might not be that much of a decline.
We had to move my DGM from one care home she was in when her dementia advanced and the lovely care home she was in could no longer meet her needs. She did decline after the move, but she was declining anyway, but there was a big jump (she never walked again), but it didn't seem to bother her emotionally and she lived another 4 years after moving her.
I do understand that when you are paying those sums of money you expect care to be amazing, or at least good enough to keep track of belongings etc but unfortunately all care homes do struggle with keeping track of residents things and the majority of the time that is not intentional, it is just because of the sheer volume of things they deal with. I made sure everything of my DGM's was very clearly labelled, but still some things went astray (never anything of any value though), although both her care home would search for things when asked, and 95% of the time things did turn up.
If you do not feel the care he is receiving is good enough then yes move him. If that is to a cheaper care home then great, but base your decision on the care component.
Also just in case you don't know about this does he already recieve a nursing care benefit? My DGM's care home sorted it out for us but back in the mid 2010's she was entitled to about £700 a month which the nursing home claimed and then knocked off our bill. Also if you haven't already read up about NHS Continuing Health Care as at some point his funding may be able to come under that. Again my DGM's care home applied for that on her behalf when she met the criteria. Also once his savings are below a certain amount (it used to be 27 and a half thousand) then you can apply for social care funding which is done on a sliding scale.