I've read the myth about renewing US licences for 12 months by re-entering the country (though I know people who do it, anyway), but I'm curious as to how the DVLA would verify this information if I applied for a provisional licence?
DVLA would neither know nor care, because you can apply for a provisional license whether you're still within the 12-month grace period or not (and whether you even have an American license or not in the first place). Being issued with a provisional license in no way affects your 12-month entitlement to drive.
It would be the police who would consider reporting for a summons if you were stopped and they found reason to believe you were beyond your year's allowance.
DVLA's involvement with your American license extends only as far as creating a "ghost" license record here if you are convicted of motoring offenses which carry penalty points, so that those points will automatically be added if and when you get a British license.