Correction: TFA says they can apply for REIMBURSEMENT. I don't know about you, but I don't want to pay $5000 now and then wait several months to get my money back.
As opposed to TFA, the actual law says:
Where registrar investigates (4) If the registrar is of the opinion that an investigation under subsection (1) would impose an undue financial burden on the land owner, the registrar shall undertake the investigation. 2002, c. 33, s. 96 (4).
Nowhere in the law does it say they have to apply for reimbursement. They can challenge the order to hire the archaeologist by stating they don't have the funds to hire him.