What extra leverage do they have to influence the occupant?

I would open this discussion up with their broker to navigate through the legalities & influencing factors of each option.

Lenders will not overturn the opinion of an appraiser, they may not influence the appraiser's opinion in any way.What you can do is to ask for a review appraiser's opinion, you can also file a complaint with the AIA as to the practice of making the adjustment. 

How much, if at all, should this influence the purchase price or decision to buy....?

He was part of my influence.

I wonder if all of the mentions are improving my influence score..

I'm old school, I believe that RE is a personal business, that personal contact is important, but I can also see that your comfort level and skill set may influence how you work, but I just prefer to answer the phone and get moving on the matter.

That's why one of my favorite Books / Audio program is "How To Win Friends & Influence People".

Most of those are in the top of influence pool and well respected as well as much more experienced than you.

The only reason I mention it is that this is the way you get negative influence, and if you're trying to network, you surely don't want that.