Local authorities do things differently, so do care homes, so it's difficult to give you a definitive answer.
In Mum's case I received a letter from the council finance unit informing me of how much Mum has to contribute, confirming that a deferred payment has been agreed, and stating that the invoices will come directly from the home. Mum's contribution has to be paid but I'm not paying anybody anything until I receive an invoice. If it comes form the home I will pay them and if it comes from the council I will pay them, and will have an audit trail to show how much has been paid, when and who to. I don't see how you can pay anyone until you receive an invoice?
With regards the care home contract, the local authority advised me not to sign it as they have agreed to a deferred payment so the home contract is with them, not Mum/me. You say that your Mum will 'eventually be fully funded by the council'. Have they now confirmed (in writing) that they will give you a deferred payment? If so, inform the care home of this and don't sign the contract.
With regards to needing to be a POA or COP to sign the care home contract, I don't think there is any legal requirement for this - you will need to look at the wording of the contract. Mum's care home contract has various signing options. The resident can sign, or if the resident lacks capacity a 'sponsor' can sign in their best interests, (providing that there has been a best interests decision meeting), or someone with lasting power of attorney can sign. The contract refers to an application to appoint a COP deputy taking approx. 6 months, and in such a case 'sponsors' can sign the contract. As previously advised by myself and others here, don't sign the contract until the financial position has been finalised.
It's not good that the care home are pressurising you to sign the contract if the deferred payment has not yet been agreed. Mum's care home have been fine. They sent me the contract months ago but haven't chased it up. It seems odd that the care home wouldn't be used to this sort of situation as your Mum can't be the only resident who has been in this position.
As mentioned before, I would explain to the care home that the council have not yet finalised the financial assessment and you can not sign anything until then. In my case, when the care home manager asked what was happening regarding payment I simply sent an email to the financial assessment unit, copying in Mum's social worker and the home manager, advising that the home had been asking about the finances. That way I had an audit trail and the home manager and the financial unit got on with discussing the finances themselves without any involvement from me. The situation was stressful enough without getting involved in discussions about who was going to pay what so I stepped back and left it to them to sort out. I can only suggest (again) that you do the same.