The way you find out what the bank is willing to accept is to have your agent call the listing agent and ask the question. Short of a different answer there, it is assumed that the bank will accept that asking price of the property.
Lender auction: Call a local title company and find out who registered the mortgage. This won't be free, but it will save you from having to show up in Los Angeles and sifting through documents yourself at the county courthouse. Most lenders are willing to speak to anyone who is willing to make them an offer. However, most have agents who handle these things for them, so the questions should go through the agent.
Tax lien sales on good properties are few and far in between. There is no magic to making money in real estate. The only way to get it done is good advice from a reputable agent and hard work looking for good deals.
The banks have their own formulas to determine what they are willing to accept. The formulas are as different as the houses they own. Just keep making offers until one is accepted. I hope that this helps