The letter should be addressed through the gallery to protect his privacy. These days, anyone can call himself an appraiser, from the framer or restorer who appraises high to justify his charges, to the antique dealer who knows a little about everything or to a potential buyer who appraises how to reduce the price. Picking an appraiser is like picking a painter or a conservator. Get a listed professional who is familiar with the type of art you have, and do tell him the purpose of the evaluation. Price is the appraisal put to proof.
Price is what the work of art brings at an auction, for example, compared to the pre-sale estimate.
At a gallery, price is the verification of the dealer's asking price that was based on his appraisal of what the market would bear. The best piece to buy is unique and available just once in all likelihood. You should not let it go for a modest difference in price that is still within your budget. The price will take care of itself, although you may have to wait a year or two to correct the overpayment. What you buy takes precedence over the way that you do it.