Basically, the listing broker has a fiduciary responsibility to get the best and highest offer s/he can for that property, and the banks usually won't budge much on the price for a while--until a property has been on the market for at least 120 days. Tenacity and persistence are key here; you have to wait them out, and if the broker asks you a few more times to send him/her your best offer, then resubmit your offer. S/he will get the point after a while, and so will the REO manager handling that property. Also, as you stated earlier, if the REO manager really had at least 1 higher, all-cash offer on his/her desk, then you probably wouldn't have heard back from the listing agent, because most loss mitigation departments (typically the department in most banks that handle REOs, foreclosures, and pre-foreclosures) are already severely understaffed and overwhelmed by their caseloads.
Another thing to keep in mind is that negotiating on REOs isn't for the faint of heart; rather, it's a long chess match. Just make sure you keep in constant contact with that listing broker until: 1) the bank accepts your or another offer, or 2) the bank opts to re-list that property with another agent. Meanwhile, maybe you could do some digging around for information on the REO manager handling that property, and perhaps you could reach out to him/her directly.