I think they are trying to play hardball with you. If you agreement does not state Conventional Financiing, then you could hold them in breach. Yo've fulfilled your end of the agreement.
One thing to check is in the definitions page if there is one which may state that financing must be conventional but somehow I doubt it.
Let them know that your next step will be to contact your lawyer and also seek damages. You don't actually need a lawyer - just say you have one.

