I'm a little late to this discussion, but I have a few things to add.
First, charcoal in a gas grill is not a good idea. It's not a question of if the ash will clog up your burners, but when. Sure, if your burners do get clogged, you can clean them, but it will be a major hassle to do so. Charcoal also represents very intense localized heat. A gas grill isn't manufactured with this kind of heat in mind. You could end up warping the grate or, if the coals are near a wall, you could warp the wall.
Even if you can keep ash out of the burners and prevent damage to the grill, as has been mentioned, the extra heat from the charcoal is the wrong kind of heat, since fast baked pizza requires intense top heat. Even if you put the charcoal to the side of the pizza, with the way the charcoal burns, and the height of the ceiling, you still won't get the top heat that you need for a balanced fast bake.
Your average gas grill, out of the box, is a horrible tool when it comes to fast baked pizza. You've got the top/bottom heat balance issue and the peak temp problem. An insert can resolve the top/bottom heat balance, but, you're still severely handicapped by the grill's peak temp. Not to mention, your average insert is typically going to cost you about half of what you'd pay for a real, Neapolitan capable, propane pizza oven. If someone wanted to match their home oven results with a grill, sure, a $150+ insert is a perfectly fine option (I'd probably recommend this one here). But 600 degree pizza is obviously not what you're looking for. If you want fast baked pizza, you want the right tool for the job. $300 will get you an Ooni Koda (gas). You don't want the Karu or the Ooni 3, since the Karu is still too untested, and the pellets of the 3 are a huge, messy hassle. I would also dissuade you from the Ooni Pro, since the thermodynamics of the larger Pro don't seem to be on par with the smaller Koda.
If your pockets are considerably deeper, the Pizza Party Ardore is a step up.
The pricing for the Ardore has been all over the map. At one point, it was as low as $600 shipped, but then it crept up to around $1100, and now it's showing a waiting list without any price listed. The Ardore has three advantages. First, a Koda can do authentic 60 second Neapolitan pizza, but an Ardore has the BTUs to do 45 second bakes. 45 second pizza is very advanced, since, at that bake time, it's pretty easy to end up with raw dough in the middle of your crust. If you do it right, though, 45 second pizza can be pretty breathtaking (45 second naan is also otherworldly). The Ardore's second advantage is it's real estate. With a 15.75" x 15.75" stone, it's exponentially easier to turn the pizza in the oven (with a good turning peel). Lastly, the Ardore puts the burner on the side, rather than on the back like the Koda. Having the burner on the side allows you to watch the pizza as it bakes and turn it a bit more effectively.
45 second pizza is not really a dragon that many folks are chasing, and with a sizable learning curve, you can master turning in a Koda. At $600, I think an Ardore is a viable upgrade for the truly obsessed, but, I think $1100 for an Ardore is a bit steep. It depends on how deep your pockets are.
If you want to compare some of the specs for the popular brands of outdoor gas ovens, I compiled a spreadsheet here:
https://docs.google.com/spreadsheets/d/1RkK7rmQMJWUYxp0zHhVLCcjQ1cLIJpUuTMOaOr2iEDk/edit#gid=0
