It's villain in the BB?

What was villain's turn action?

I guess, he checked.

As for the action, only reason for betting would be protection. "Protection" in and off itself is not existent as a reason to bet, so it should be +EV (compared to xb). It would be +EV, (for instance) if Villain would always fold (no x/r, no river-bluffing).

In all other cases (Villain is waiting for a x/r, Villain x/c with a fraction of made hands, Villain would bluff the river if a brick / overcard comes), it's most likely better to xb. We preserve our EQ against the threat of a x/r, Villain only has ~25% EQ (with overcards) and we have a great bluffcatcher for the river. Summarized there's too little value in betting.

But in reality it's likely close. If Villain is prone to cbetting (with most of his missed overcard range) and is one-and-done after getting called, it's probably better again to bet (for the reasons mentioned).

I'm raising the flop. 500. Maybe a little bigger actually.
As played, I am betting the turn. About tree fiddy sounds good, maybe 400.

Raise flop often.

Always bet turn. You need protection from overcards and you have great equity vs a checkraise. The way I approach deep spots like this is to bet the largest sizing that makes it very uncomfortable for an overpair to checkraise all in. So if pot is around $620 here and you have around $1800 behind, I'd bet around $240 and build my range such that QQ-AA are near indifferent between X/shove and X/call turn. So bet something like JT+ for value sometimes and then mostly fold QT-AT vs XR. Then bet some random medium stuff occasionally, but more polar, like bet 66 occasionally and then jam on good rivers. Bet a lot of bluffs which you have a ton of. KJ-KQ, A8s AJs etc etc lots of combos, then follow through with those as your range permits on rivers. Tend to check behind the FD combos that lose a lot of EV against a XR; for example if you have Ad5d I'd often X behind as a XRing overpair will force you to bet/fold which is a disaster.