I mean, maybe. But doesn't St. Louis have a pre-arb, Gold Glove middle infielder in Tommy Edman who has already been worth 5 rWAR this season? And DeJong cheaply under contract for a few seasons? They have Arenado at 3B, a team OPS+ of 117 and a team ERA+ of 99.
If I'm running the Cardinals and I have budget for an FA splash, I'm looking for starting pitching (and maybe an outfielder?), aiming to trade DeJong, and starting Edman and Gorman up the middle. If they're signing Red Sox FA to be, Eovaldi on a 2/$40-ish deal would be a pretty good fit for them, honestly. You'd have Wainwright (if he's back), Eovaldi, Mikolas, Montgomery, Hudson, with Liberatore and Gallegos as high-upside depth in AAA.
All of which is to say, I don't think Bogaerts really fits their roster situation well enough to be worth beating a presumptive Boston offer. If his market really collapses, that's another matter. But if his market looks like it could be bad, he'd opt in to his 3/$60+option with Boston, right?