my mum had a similar issue. she was working as a supply teacher through an agency and then took a part time job at a school she supplied at (which she also used to work full time for about 10 years ago). agency went mad and demanded money from the school. the school turned around and said they wouldn’t pay and would look to use other agencies to supply cover staff in the future at which point my mum’s agency got very, very apologetic.
just hire the person and tell the agency it’s up to them if they want to do business with you any more. you’re in the driving seat, there are other agencies, but i bet that agency relies more on you than you do on them.