I have been close to a similar experience very recently. Absolutely no judgment from me, but I would very gently suggest that yabu (I was told this myself on MN at an earlier stage, and came to realise it was true). The, reality is not only that looking after someone with dementia full time is incredibly draining, but also that not everyone feels equally suited to doing it. It sounds like there are many things you could quite legitimately criticise your DS for, but I don't think this is one of them. She may feel unable to care for your mum properly - or she simply may not want to - and either of those should be fine. Even allowing for the fact that a child may have 'received' a lot from their parent (in terms of money, support etc), it still shouldn't be an expectation that they provide full time dementia care in return, unless they genuinely want to do it.
The reality (as I was told myself on these pages), is that however much you feel that your mum isn't ready to go into a home, you can't push for that decision unless you're prepared to take on her care yourself full time. You may well feel very sad about that - it's natural - but you can only make decisions in as far as they affect you - not others. You may feel that being at home is the best thing for your mum - and you may be right - but even if you decide that your mum's needs trump your own, you can't make that decision on behalf of others in your family. So, even your 50% idea is unreasonable, because it still compels your DS to take on 50% of the care, which it sounds like she doesn't want to do. The question is not 'is mum ready to go into a home', but 'is there a way of having mum living at home which doesn't impinge on the freedoms and wishes of others' - and it sounds like the answer is no.
The other thing I would say (again, based on experience), is that even if staying in her own home is in theory the best thing for your mum, living in her own home under the care of someone who doesn't want to provide that care, and is doing so for reasons of guilt or obligation or self interest (as it sounds like your DS may already be doing) is very unlikely to in fact be the best thing for her - and may lead to a very poor standard of care.
Having said all that, is staying at home with live in carers an option? Private live in carers would be able to take her out as much as she feels able. But you say she's resistant to carers..?