I wouldn't stress out about the pre-approval letter. They aren't legally necessary, and mainly exist as a reassurance to the seller that you aren't wasting their time. Once you nail down a house and an approximate purchase price, you can ask your lender to issue a mortgage commitment letter, which is slightly stronger than a pre-approval letter anyway. As to your question of how much to have the mortgage commitment letter written for, it depends.
In general, if the house has been on the market for a while and you suspect (or know) that you are the only current offer, I'd lean towards a lower number. This will enable you to pretend to be forced to cap your offer, which could help prevent a higher counter-offer from the seller. However, if you expect (or know) there will be multiple offers, a higher number may be better, because from the seller's point of view it seems to lessen the chances of financing falling through, and they may be more likely to select your offer.
That being said, note that commitment letters are for the mortgage amount, and may not consider that the down payment amount can vary as well. The fact that you can simply put an addition $10K down if you want to enables you to increase your offer without getting a new commitment letter issued every time.