I had similar problems with my partner, who had Vascular Dementia. We only had an over bath shower, and it simply became too dangerous to try and get in to climb in and out the bath, and uncomfortable and frightening, (I think for him). He would let me wash his hair ( you can get no rinse shampoos which seem to work quite well, although hair does seem to 'self clean' after a while - I only wash my mum's about every few months and her hair always seems clean - Dad's, though is another matter). He would let me shave him, he would let chiropodists cut his toenails, teeth cleaning was a problem (constant nagging from me) but he would use mouthwash without prompting, and every so often I would sit him on a chair in front of washbasin ("Right - just a quick wash today love" - "Can we do it tomorrow?" - "No, let's just get it over and done"), give him a soapy flannel (just wash your bits, pet), and I would do arms, legs, chest, back, etc., with flannel or wet wipes, etc. You can get no rinse foam, which can also be useful.
I am well aware that not everyone is as lovely as my man was, and just will not allow any of this sort of thing. It wasn't always easy, and sometimes you have to pick your moments or just let it go for a few days. Some PWD will respond better to a carer doing this, and certainly "the doctor says . . ." is always worth a try. I used to tell him we needed to make sure he didn't get any skin problems, because then we would have to see the doctor.
He would put on the clean clothes I put out for him, and never smelt (that I was aware of, anyway!), and I probably only managed the wash about every 3-4 weeks (I long ago gave up worrying about the OTs going on about bathing/showering every day - they clearly had no idea what dementia can be like!).
All I can say is keep trying various ways, try not to stress about it too much, and things do change so maybe what seems impossible today may become possible in the future.