I am not sure why it's so important to you, but since I am that way about health information, I'll just go with it. However, before I do, let me tell you that lenders provide confidential information often to me - not REALLY confidential, but basics, and it STAYS with me. No exception, no matter what... no matter who I am friends with. I take my job seriously. AND, I use the information ONLY to help... and sometimes it does help that I have some of that information - I can offer alternatives to the lender or borrower or clarity on something.
Having said that, just tell the lender that the agent is on a need to know basis and that other than the approval, you want them not to discuss anything with your agent. If there is a specific reason that the agent needs to know more, tell the lender that you'll need to specifically authorize that information to be shared. The lender should honor that.
Lastly, I wish to address the idea of you not using a lender suggested by your agent. Right now, I have a buyer who may be in the midst of a lawsuit because he insisted on choosing his own lender. With a down payment of over 50% of the cost of the home, and 60 days to get the loan approved, settlement came and went without full approval. I have no weight with his lender. Neither does he. MY lenders would NEVER do that. First, my lenders became my lenders because of careful screening and repeatedly showing me what they could do - and doing it well. I refer them because I have the comfort level of knowing they will deliver. An internet lender? Not so much. Someone you used 10 years ago? Not so much. Trust me on this. All lenders are not created equal. Plus, if they screw up, I would never provide them another referral... and I would talk with every agent I know and tell them about the experience, and I'd blog about it... and trust me, it would hurt that lender. So they have to treat every referral extremely carefully.
So if you want to pick your own lender, it might work... but it might not, and you're at their mercy. PLUS, you are held accountable for the decision of choosing your lender. Your right = your responsibility. So, they don't close or close on time - you're responsible.