I apologise for the delay in any expert responding to you.
This is an unusual area of law.
The way that the care home regime works is that the care homes will enter in to a contract with local authority for a particular resident. They will agree a price which is not usually the rate sought by the care home but based on the resources available to the local authority to fund it.
Where a resident has some element of medical need the NHS will pay for part of the care home fees depending on the level of need.
What has happened here is that the care home have realised that they cannot continue based on the rates agreed with the local authority. Presumably they were agreed by the previous owners. They have put the prices up but the local authority do not the resources to increase what they are paying.
It is clear that there are affordable alternatives, hence the plan to move your Mum, so it is not clear why this care home cannot be viable on the same income. However, care homes are struggling to survive and they are closing at an alarming rate.
As such the bot***** *****ne is that you and your Mum are caught between the squeezing of the fees paid to care homes by local authorities who are having their budgets cut massively each year by central government. I appreciate this does nothing to help but does explain why this is happening and that you are not alone in this situation.
There is, however, nothing that can realistically be done, the care home cannot be forced to continue to run at the rates agreed as it can simply close down.