I offered $45,000Bank countered at $47,000I countered at $46,000They accepted!

They accepted the offer, but had an HVAC contractor provide a quote on the entire system and instead agreed to write us a check instead so we can replace the system ourselves.  

The lender will not accept second position liens usually.

Later today, the seller accepted an offer from someone else. 

Since she went through bankruptcy, I have no idea what kind of places would accept someone with a bankruptcy that doesn't have a co-signer.

If you're not, there are still several 506b syndications out there that accept non-accredited, you just have to add that to your search criteria.If you need any help with this process or are looking for referrals to start the process, I'm happy to help. 

This would potentially bring the price point into my acceptable range.

If you think that means they won't accept it, that doesn't matter.

I think I might have a few people in that post-bidding price range, I'll let their Realtors know if it comes up" is broadly accepted as normal activity (I'm not commenting on what SHOULD be, merely on what IS) unless some sort of confidentiality is specifically requested. 

You either want a more personal approach and accept that it comes at the cost of flexibility or you choose the big guys and forgo the personal touch.