You are negotiating with the seller. Before any offer goes to the bank the seller has to approve it.
1) You can offer a less amount, however if the bank did not accept you first offer they likely will not accept an even lower offer. In fact the owner may not even let it be presented. If the seller has a good agent that agent will advise against presenting your new lower offer. Time is of the essence for distressed homeowner. As a Certified Distressed Property Expert, I would never let my client tie up the property by presenting an offer that is very unlikely to be accepted. The reason you likely waited so long for a counter is that the bank was ordering a Broker Price Opinion (BPO). BPOs are prepared for every home a bank is going to sell short, or foreclose on. A BPO is an independent price analysis from a 3rd party Realtor. Banks rely heavily on this price opinion in their dealings.
2) When the bank countered your offer, they rejected your original offer. That offer is now officially off the table. You are not obligated in any way to buy the home at the $291K and should be entitled to your deposit back.
Equally as important as your questions was one of your comments though, do you feel your agent isn't "on your side" because he, or she, isn't telling you what you want to hear, or do you really not trust them. If you feel your REA isn't "on your side" because you think they are untrustworthy find a new agent. Why would you work with someone who you don't feel has your best interests in heart? However before you do that, stop and take an objective look. Do you think they are just not on your side because they are telling you you should be offering more? Perhaps you should be? I can't say without knowing the details of your purchase, I just suggest you take an objective look and know the difference between a shady untrustworthy agent and one who may be telling you the truth, even if the truth isn't what you want to hear.
Sales Associate, CDPE
RE/MAX Gold Coast Realty
56 Newark Street
Hoboken, NJ 07030
Direct: 201 795-5200 x340