As Pete R said the LA have to find at least one placement that meets her assessed care needs, whether this costs more than their tariff is irrelevant, they cannot force you to pay top ups.
If you go down the top up route make sure that you pay the tops to the LA not to the home, do not sign anything with the home.
Remember home rates are likely to rise at a faster rate than the LA tariff increasing the top up demanded.
If the only home the LA can find would mean that they have to pay almost as much as the current home your hand for keeping her there is strengthened. Your argument would be that it is against her best interests to move, she is settled and knows the staff and they know her, see if any medical staff will back you up on this.
Insist that the LA nominate at least one home without a top up, look at that home and take it from there.
If they argue about the fact that they have to offer at least one home and if you choose that home they must not ask for a top up remind them of their duty under the Care Act
Page 397
Sec 12 of Annexe A of the Statutory Guidance of the Care Act 2014
A person must not be asked to pay a ‘top-up’ towards the cost of their accommodation because of market inadequacies or commissioning failures and must ensure there is a genuine choice. The local authority therefore must ensure that at least one option is available that is affordable within a person’s personal budget and should ensure that there is more than one. If no preference has been expressed and no suitable accommodation is available at the amount identified in a personal budget, the local authority must arrange care in a more expensive setting and adjust the budget accordingly to ensure that needs are met. In such circumstances, the local authority must not ask for the payment of a ‘top-up’ fee. Only when a person has chosen a more expensive accommodation can a ‘top-up’ payment be sought.
https://www.gov.uk/government/uploa.../file/366104/43380_23902777_Care_Act_Book.pdf