DJ, the thickness on the Lodge pan is deceiving because the lip is larger. I have yet to know anyone that has taken a caliper to one. It might be 3/16, but I definitely don't think that it's 1/4."
I fully understand your concerns regarding size and applaud your inclination to go larger than the 14" lodge pan.
Jeff's doughnut/break idea is worth trying, but that kind of approach can be hit or miss. If you can befriend a metal guy- the guy actually working with the metal, then, yeah, you might definitely be able to walk away with a piece for practically nothing. The moment you start dealing with the front office, though, you'll most likely be shelling out 20 just for the cut.
Jeff has expressed some safety concerns regarding junkyard steel, but if I were shopping for steel and wanted the best possible price, that's probably where I'd go. I'd treat it like hazardous waste until I got it home, but after a good layer of naval jelly, I would totally eat off of it.
Since you're not really all that concerned about bake times, I would look into unglazed tiles. They tend to be incredibly difficult to track down, but it wouldn't hurt making a few calls just to make sure no one local carries them. If you get lucky and someone does carry them, you could have a $10 solution that should be comparable to 1/8" steel, and, depending on the density of the tile, might even match up with 1/4."
I googled tiles Phoenix and came up with Home Depot and Tiles for Less. I'm almost positive that Tiles for Less will carry them, but the question will be whether or not they sell them individually, as an entire case could get pricey.