On my trailer, I don't think there's any way to get to the top of the wood beam to perform a bolt, plate and nut fix. But I agree this would be the best kind of fix.
If I had this problem, I think I'd try filling the hole with epoxy, or maybe Bondo, then reinserting the lag screw, before it cures. There is a type of Bondo that's designed to adhere to wood, but maybe it would not adhere to the metal of the screw. However, it would mold to the thread pattern and should provide a firm grip.
Hi-Lo 1707T - Tire Minder TPMS on Tow Vehicle and Trailer, 300W Solar Battery Charger, Equal-i-zer WDH
2014 F150 Platinum 4x4 3.5L EcoBoost SCrew