Andrew, don't drink the kool-aid!
Because cordierite stones (and steel) work so well for other people, it's hard not to look at them and make the assumption that they're inherently good and that they can make great pizza in any environment. They're great for some people, but they're horrible for others. You're one of those unlucky others. For a 4 minute bake, you'll most likely have to pre-heat your cordierite to 575. Without a top heat source, that means that you're pre-heating your ceiling to 575 as well. 575 for a ceiling is nothing. If you forgo a false ceiling and use the almost massless ceiling of the oven, you're talking about almost no top browning, and, if you pre-heat the false ceiling of tiles to 575, you're still talking about a completely unbalanced top to bottom heat ratio. The only way to get the right balance of top to bottom heat is to use a hearth material with a low heat transfer rate, a material that will give you a 4 minute pie at 650 or higher. When you can pre-heat your false ceiling to 650 (or better yet, 700), that's when you'll start seeing proper top browning in 4 minutes- that's when the magic happens.
Because of a bottom heat source, you're extremely limited to the materials you can work with, and have to choose those with the heat transfer rates that give you the proper top to bottom balance. If, like steel is a relative super conductor of heat, there was a material that was a super emitter (to give you intense top browning at low temps), then you could have the freedom to use a greater variety of hearth materials. Unfortunately, emission is what it is. There's no magic emission bullet. Thermal mass in the ceiling buys you a little better emission, as does a darker colored surface, but, at the end of the day, your pizzas will only be as good as the temp you can pre-heat your ceiling to, and, when working with a cordierite hearth, that temp is too low.
You can get an 18 x 18 x 1 cordierite stone and probably get a relatively evenly browned, cheese heavily melted, 8 minute NY pie, but it won't touch your current broiler pie. The setup I'm outlining, if you can put it together, should give you something both a bit better than your broiler version and a lot easier to make. Instead of all the kneeling and the hassle, think launch, maybe turn once, 4 minutes, perfection (or close to thereabouts
It just occurred to me that my steel bar idea works with a single stone, but won't work for tiles, especially not smaller ones. Perhaps you could put the tiles on a cookie sheet, but you'd need to make sure to use a gauge of metal that doesn't warp.
From a biggest stone possible goal, that 2" gap in the back is kind of depressing. I hate you lip!
Don't give up on the quarry tiles. Bring back the broken ones, get your money back and start looking for an actual tile store. They should have a greater variety of tiles than HD. For now, size the tile hearth to the 16" dimension (grrrr). I want you to get the tile hearth and ceiling in there (foil is fine for covering the gap in the ceiling, just make sure you leave some room for air flow), fire it up and see what it can do, first without a condom probe, then with. Once we can see exactly how much top browning you can get with a 650/700ish preheated close proximity black glazed tile ceiling, then we can see about stepping up to an 18 x 18 stone (possibly fibrament).
I just talked with Chickenparm (Bill) about the 3/4" fibrament stone that he recently purchased and he's getting good 4 minute undercrust char in the 575-600 realm. As previously discussed, a 1/2" stone will push that higher, maybe as high as 650, but I'd still like to see you with a material that will do 4 minute bakes in the 700 realm. It's a shame they don't do 1/4" fibrament, but I'm guessing it would be pretty fragile. Worse comes to worse, you could use 1/2" fibrament with an 18" screen. The screen will temper the transfer and allow for 700ish bakes.