Looking for advice,
Typically the exclusive buyer agency agreement has a fee in it usually 3% of the sales price "purchase price". Is that not in there? They just wrote in MLS fee? This is kind of a sticky area because you are contracted to work with that agent only and buy from them. The seller of the private sale can pay the fee of 3% or they can try to negotiate it to a different fee, if that is the case then you need to get in writing from the agent you have contracted with that they accept that and nothing further. You really need to consult with your Realtor or a competent attorney, there are many questions that could be asked here.
What is the term of the agreement there must be a start and end date and even more...my best advise is talk to your agent and see if they can work something out with the owner of the private sale, or consult with an attorney about the contract between you and the agent.
Best of luck and best regards