You'll still have the up front purchase price but yes, with a half lease you'll share at least a portion of the monthly board. Just be clear upfront about what the lease covers. What happens if there is an an injury especially any injury that takes away riding time as she would be effected in that case. As a minor your parents will need to have a lease agreement drawn up and signed by the other girl's parents. They may want to talk to your Aunt if she leases any of hers or your instructor to come up with something fair and within reason for the lease agreement.
Some horse people change their horse, they change their tack and discipline, they change their instructor; they never change themselves.