It comes down to who will give in first. Obviously, the buyer has nothing to lose since their money is already tied up, so the buyer probably won't give in and release the earnest money to the bank. This is what should happen:
Speak with a real estate lawyer and your agent about your options. Make sure all aspects of the contract have been reviewed by the real estate lawyer, including any addendums. They can give you an idea of who is right.
Buyer sends the release of earnest money and demand letter to the title company and CCs the bank, stating the reason that they feel entitled to have their earnest money released OR the bank agrees to provide the documentation needed for the loan and extends the closing date (if they still want the house).
If the bank refuses to release the earnest money or extend the contract, at that point you send them a notification that you want to go to mediation.
Here is the key point to this: The earnest money is effectively a lean against the title of the home. While there is a lean against the title, the bank CANNOT sell the home. Depending on how long it takes to go through mediation, often times the bank will concede and release the option money in order to get a clear title.
What confuses me though is the fact that the bank is not wanting to extend the contract on the home when you have financing and only need the termite report in order to close it. Something doesn't sound right in that scenario.
Best of luck regardless of the outcome.