"up until the point the bid exceeds the bid price" What ever does that mean? The bid is the bid price !
As you do not know what an auction might go to, it is up to the seller whether to add "Best Offer".
If you accept a Best Offer you might be losing out and if you decline a best offer you also could lose out. You make the choice, but once a bid is placed you have agreed to sell to the highest bidder, you can't say "hard luck" I am selling to someone who has made an offer. You have no idea how much the bidder has bid and you will never find out.