I'm on record upthread that the deciding factor--assuming the bids are in the same ballpark--will be where this young man wants to spend the rest of his 20s and early 30s, so location over all.
But if I were making the offer for the Sox, I would lean into the Treat Williams/Michael Ovitz pitch to John Michael Higgins/David Letterman from HBO's The Late Show (Night Wars?), which is that if Yamamoto comes here, he would be the Sox signature star, and that has a value beyond measure. He is the opening starter, he's the guy on the yearbook, he's the face of the franchise. Because that is the one thing that the MFYs, the Dodgers, the Phillies, and the Mets cannot offer.
If he goes to the MFYs, he's never passing Judge, and now they have Soto, and that Martian kid coming. They already have the Cy Young winner in house, ask anyone how much of an event any of Cole's starts are there. (They're not.) Because the MFY fans, for all their team's success, have never prized their pitchers over their sluggers, which is why the team has never in their history have had a pitcher as their signature star. Not Waite Hoyt, not Lefty Gomez, not Allie Reynolds, not Whitey Ford, not Ron Guidry, not Andy Pettitte, not Roger Clemens, and not Gerrit Cole. (I'm old enough to remember Guidry in 78, but that was a blip, and he was still behind Thurman and Reggie.) They had the greatest relief pitcher in the history of the game, the guy who was arguably more responsible for their last great run than anyone, and even he played second fiddle to a shortstop with the personality of a carp.
Compare that to the Sox, who have a long history of celebrating pitchers going back to Cy Young himself. And if he has any doubts about that, we can remind him that when we had the best player who ever played the game, we prized him for his skill as a left-handed ace pitcher, while when they got a hold of him, he had to bash a bunch of home runs to get any attention there.
(I know it's silly and desperate, but that's where we're at right now.)