I guess many carers have joint Bank accounts with those that lack capacity to manage and even understand their own finances especially with new technology apps and websites which the elderly see as on another planet.
The old way was having a verbal permission to use the account for the benefit of the cared for person however recently it is now being interpreted by companies and firms that it’s open to abuse and you need to prove through transactions it is for the benefit for the cared for.
The proper way is a power of attorney which some elderly may refuse to give to one caregiver even family, there is also a 3rd party access mandate through your bank however the account has to be a sole account on the cared for name only for you to have access through the legal offical route.
It’s not enough you need a formal arrangements of finances with the person you care for they ask copies of last 3 months bank statements all bank accounts and if they see say a pension on there they assume it’s you who has the pension that you did not declare on your intitial application it’s open to misinterpretation and can lead to court actions to justify your position.
Have you thought about asking the LA for a “Care Act Advocate” someone to support you through this? It sounds like they are being very difficult over all this, an advocate might make them more polite? You can’t produce something you don’t have! I would be inclined to request a copy of their policy and procedures when dealing with things like this, as a Freedom of Information Act Request, made formally via the LA’s website, saying it’s urgent.