While this may seem like a no-brainer, they may hope to get a higher bid closer to market value, and so reject my offer and proceed to foreclose.Alternatively, I can let them foreclose and I will lose the property, but somehow have them pay me whatever i paid for it?

learn about 1031 tax deferred exchangescomplete you financial package - 3 years tax returns P&L - real estate portfolio, credit scoreexpect to be rejected by lenders and other W-2 employees - qualify lenders, bankers, mortgage brokers - ask what products they have - see if you qualify for themlook at 20 buildings before offering on any - know the SF dollar cost - keep a journalmail lots of letters (be professional NO yellow childish communications) to owners of apartment buildingsUnderstand what a CAP rate is and how to get to it - fill out your  commercial APODVerify -verify - verify all financial data given you by sellers - sellers are liars (so are buyers) trust NO One!

She can accept the offer, counter with her own, or reject the offer.

You have many options to justify rejecting.

Expect that it will likely be rejected and move to the next.

This will often result in me having to reject their application without going any farther or wasting any more time.

Always contact their previous landlord, if they don't have one reject their applicayion.I place a great deal of emphases on credit score since it is a measurement of financial responsibility and financial planning.

What specifically are the people saying to you when they reject you? 

That way I can find a lender that doesn't have overlays that would reject my loan application.

When they call or email ask specific questions regarding the items in the add and make it clear not meeting those requirement will result in a rejection.