Yeah, as long as he's not "impossible" you should be able to get him if you pay enough. Does he have other options? IOW if he's a college senior not may places to go other than sitting out a year. Unless you have independent leagues? I don't know on that as I don't have ILs. If he has the option to return to school than he has more power.
In either case, if you know he's a hard sell, then overpay on the first offer if you can and get the contract signed. How much? Yeah that is the question
I don't think the large ask is because they don't want to sign with a weaker team. As IRL there aren't a lot of options, players almost never go from school to the show, so it's more about knowing their worth in the market and using what leverage they have. Again an underclassmen or HS player has quite a bit of leverage.
The large ask can be to force teams to not draft them. Think of Mark Prior and his price when the Twins picked 1 and the Cubs 2. However in that case the Twins got Joe Mauer and who's to say they wouldn't have selected him anyway over Prior? But in any case those large amounts are there, like real life, so teams do have to take into account if they can afford to sign the guy. If they can't the may have to be happy taking the best guy available they can sign.
But then back to the original point on changing the offer? Not happening, you'll have to wait for the reaction. That goes back to my earlier point of making a "great" offer for a first offer if you really want/need to get the guy under contract.