The process to get your daughter the British passport is relatively easy as long as you have the required documents. You will need the C2 form and a couple of UK standard photographs of the child, hospital scan photos and receipts etc etc . It can take up to 8 weeks to get the passport returned, but in the case of my son it took only 4 weeks.
I can't remember the cost, but certainly it's going to be cheaper than applying for a different visa and she has the right to be British and all the advantages of our country