I'm shooting a Mossberg 500 12 ga and 3" #7 Hevi shot13.

I have internal Mossberg chokes Cylinder, IC, Mod, Full and X-full and extended & ported chokes of Mossberg X-Factor Turkey and a Star Dot.

What's your recommendation out of those?

I'd shoot these and see how they do. If they don't work I'd try Jebs, Indian Creek and Sumtoy, in no particular order. My guess is that one of the chokes you have will be fine at 40 yards with #7.

Star Dot shot ok in a 500 and 835 for me. Not great but good. If you decide to try something else go with IC. IC really did well with H13 in both.
Deep clean your barrel before you start testing loads/chokes. It tightens my patterns up noticeably. I’ll typically clean every 5-6 shots.

Pattern them and see. Tru Glo Strut Stopper Xtreme is cheap and a sure bet, though.

Patterning is always the final proof, but at $6 a pop I was hoping for a good starting point based on experience. I'd also like to confirm that it's OK to shoot Hevi13 through an XFull or tighter choke.

It’s fine to use any of your chokes. If you haven’t already, confirm POA/POI with some cheap shells. This will save you time and money with your premium loads.

Well, experience says the SSX choke will work with the Hevi. When your shells cost several dollars each, a matching turkey choke doesn't cost that much when it saves trial and error.

The Star Dot you have will work with 1 3/4oz Win Pb #6s. Those are a lot cheaper.

You can indeed get your sights close with dove loads before fine tuning with turkey shells, most of the time. Or use the one-shot method with just the turkey shells. The latter works as well and uses fewer shells overall.

I spend several dollars just for the shot that goes in my handloads. This is not at all necessary just to kills birds. It is an additional hobby.

I've had excellent results with lead 2 oz #4 Winchester loads. Alas, I can no longer use lead shot in California.