Now, I'm accustomed to just asking the broker up front whether he anticipates an all cash offer to save time.

Oh, the anticipation!

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You could also try hiring a PMC, but you would need to find one that can handle this situation and document every interaction with the tenant.

I also went down the 203(k) rabbit hole last night and from what I can tell, they will allow you to claim 75% of anticipated rental income but that information was just from a Google search. 400 to 500k does not get you much around here unfortunately.  

Looking forward to interacting with you all!

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Also make sure everyone you interact with knows you buy houses and what type of property you are looking for.

You may have to sweeten the deal more than originally anticipated, especially for a comparatively low rent (in my area, 1000 a month is a steal).

There's two different borrowers risk, the underlying borrower is no longer the one underwritten, and you cannot anticipate the new borrower's intentions.