This has always been my feeling about Mookie insisting on going to free agency: some place closer to home, some place warmer, some place not Boston...
Maybe I've been myopic, but to me, Betts was one of the rare baseball stars beloved by a fanatical fanbase of a large market franchise dedicated to winning who never publicly expressed a goal to remain with his original club forever. Unlike Lester, Bogaerts, Sale... even Lindor has been saying lately he wants to stay in Cleveland. Didn't Lowell take less to re-sign than go to Philly?
These guys aren't all just mercenaries. Competitive athletes want to win, and once they're set for life monetarily they start craving intrinsic rewards like Manfred's piece of metal. If Mookie sees like-minded compadres like JBJ and JD jettisoned in favor of a rebuild, that may be another factor in his decision. Look at Arenado now; a year after hitting the jackpot he's hungry and having second thoughts about staying in Colorado.
As far as signing with a team for one dollar more -- if a guy prefers one place or another, I'd think the runner-up would have to exceed the best offer by enough to make him change his mind. In other words, if he likes LA better and it offers 420, Boston may have to go to 450... but these are all hypotheticals (hypatheticals) since we already know the Sox don't deem Mookie at the very top of industry market value. Yet.