They have turned down my contingency to be able to title search the property, but they have title insurance and are willing to convey that to me.

Well, typically a financing contingency, if it doesn’t specify a max rate, usually references “market rate”.

Rehab budget: 180k (possibly 153k because of 15% contingency needed for 203k rehab)Will rent for up to $5600 per month after rehab.

You are an investor in the area and presumably could close quickly without many contingencies.

I have offers on two rental properties accepted that are contingent on financing.

We worked out a deal to get the work done for $75k, and they would get a bonus on the back-end when it sold (another $75k), contingent on timeliness and completeness.Everything went great at first.

@Ryan BreenRyan, Another strategy you could try is put a contingency clause in your offer stating:"Property to be vacant upon closing date and free and clear of all personal property" Essentially, this makes your current problem become the sellers problem.

Wrap the property up by submitting a contract with contingencies (inspection, appraisal, etc.), and then renegotiate if needed. 

After 2 months of back and forth on negotiating with the lawyer/owner we come to agreement of $260k with contingency of tenant moving out before funded the deal.

However, as an investor, I would purchase without any contingencies (yes, including waiving inspections and mortgage clauses) and therefore less muss and fuss for the owner.