inforce WMLx lights are a good fit on a shotgun but the reviews are iffy. I think the switch also lends to mounting on a handguard without unnecessary cables and remote switches. They have a good form factor, good reliability and don't break the bank. You still need to go from MLOK to Picatinny to the light but I don't know of a direct MLOK light as of yet. I have heard of battery drain issues.
The tried and true light on shotguns would be the Surefire forend... they rock, even the old ones which pop up from time to time in your price range.
For sub $200 I think you'll be looking at a used, older surefire or a streamlight.
But man I dig the form factor of a WML on a shotty.