You are right to be cautious. The care home has no authority to require payment directly from client account and paying them simply on request risks breaching Rule 3.3
Unless you have clear written authority from the attorney on behalf of the donor and are satisfied the payment is properly due and in the client’s best interests, you should not make the payment directly.
The safer approach is to complete the sale, account to the attorney and allow them to settle the care fees. If the care home require payment, they should pursue it via the attorney.
You may also wish to confirm whether there is an LPA or Deputyship in place, as this can affect how authority is evidenced.