They can't make you pay a top up and your mum isn't allowed to pay it herself.
I assume from the fact you're paying a top up that the LA are funding her care?
If so the LA have to offer at least one home that doesn't need a top up and can meet her needs but it may well not be the one you choose. If you don't accept the offer they have fulfilled their legal obligation so pretty much that's what you get, one offer.
If the LA are funding you may as well just ask them for the names of the homes that take the LA rate, they seem reluctant to do this, I can only imagine they keep them for people absolutely not pay a top up for whatever reason so the LA want to keep the cheapest places for themselves, possibly I'm being cynical.
CHC isn't given for AZ/dementia there has to be a medical condition and they are defined as being social problems so if you apply just on that basis you have very little, if any chance of getting CHC.
If it's a nursing home then you can apply for Funded Nursing Care (FNC) which is £155.05pw (down from last year's rate of £156.25), this is paid direct to the home and is there to cover the nursing aspects of her care. Some nursing homes have 2 rates, one with and one without FNC so it's as well to ask
If the LA are funding then there is a set (local) rate they'll pay, ask them what this is then see what the shortfall is and take it from there.
K
https://www.gov.uk/government/news/nhs-funded-nursing-care-rate-for-2017-to-2018