I don’t believe anyone has addressed the OP question, could ML ban the bats in batting practice etc. I imagine they could but it would be a bit odd, making an equipment rule outside the play of the game and possibly hard to enforce.
The funny bat prohibition has been one of those “wink-wink” deals. AFAIK, the league has never randomly tested the bats or anything to insure compliance. It had been several years since the last cork bat incident–it’s safe to assume illegal bats have been used in games many times in that period.