I can see how you have gone in circles and come up with this plan,OP - and I don’t think you are doing it primarily for underhand motives towards your sister as some PPs have suggested.
I agree with PPs, though, that the plan has so many holes in it that it isn’t really feasible .
The major problem for your mother atm is the ending of her current mortgage term and the unlikehood of her getting a full replacement mortgage. She certainly would have problems getting a straightforward one with an existing large ( secured ? Unsecured? ) debt that you have now recently said she will be able to resolve in the near future.
Would that mean she would have closer to the full notional £175K available for an onward downsized purchase - possibly with a small mortgage, with a small LTV ratio ?
My DM, had a similar problem when my DF died - in 60s but long retired as had been caring for him - left with a large family house that she loved and wonderful neighbours etc, but a very small pension and less capital than she had thought ( another story - financial ignorance rather than fecklessness) and unable to service the remaining, already extended, mortgage.
We were unable to offer financial help at the time ( 2 young DCs and a big mortgage ourselves) so after a period of handwringing about unfairness of life she faced up to it, sold up and bought a much, much smaller property. Much lower maintenance, much smaller garden, smaller running costs - and much nearer and just as nice neighbours. Very happy there until it had to be sold for her to go into a care home.
And there was an inheritance for me and my brother to share left over - not as much as if she had managed to keep the big house but we had never, ever assumed that there would be anything for us when she died, apart from keepsakes. It’s very dangerous, IMO , to plan based on being a potential inheritor.
You say your DM won’t contemplate downsizing but she doesn’t have to if you are offering her an escape route - one that is flawed from all angles except from hers. The house will be repossessed next year if she can’t magic up the balance due.
Equity release can be a solution to some problems but the market values they work on are very conservative, the amount they will lend is age dependent and having other loans secured on the property may also cause refusal.