You have a nice box you've invested effort in. Consider this:
1. Temporarily remove all the gas apparatus and enlarge the hole by just 3-4 inches
2. Temporarily screw a board on from the bottom that fully covers this hole
3. Screw the bottom of a 3# coffee can onto the center of this board, facing up. This will form a mold for your final inner grease/oil drain.
4. Cut a small hole in the far left or far right, whichever way you'd like grease to exit and fit a 3/4" close pipe nipple in it, sticking up.
5. Generous place steel "wire cloth" (chicken coop wire) of .5" grid across the entire bottom of your box. Make it fit tight to the circular coffee can and the pipe nipple drain. Using nuts or washers as spacers, attach it to your existing bottom so it's suspended ~.5" from the bottom. This is your "rebar".
6. Mix a half sack of Red-e-crete, just a little soupier than normal, and trowel it onto the bottom. It looks like your front lip will allow you to make it ~1" thick. Work it so the entire bottom slopes to the pipe nipple "drain".
7. As the concrete starts to "set", remove the coffee can and round the concrete edge so it's less likely to chip.
8. After at least a 24 hr cure, you can remove the bottom board of step 2.
9. Replace the underboard (that retained the concrete) with flat steel sheet (of just about any gauge) with a hole 1" dia smaller than the coffee can mold. If you don't have anything handy, a roll of steel flashing from a hardware store roofing dept can be used and attached with a staple gun. Snip the .5" annular area of steel with a tin snips in a couple dozen places radially, then fold them all in to further retain the weakly-reinforced concrete.
10. Let it cure at least a week before reinstalling the gas burner. Bring the temp up slowly (you'll always want to do this to avoid cracking your concrete) to outgas the concrete. You may want to make this burn-in a several-hour process since concrete can have an unpleasant smell.
Then you're ready for cooks up to 275 or even 300F. Beware of grease splatters on the inner wood sides. As others have said, it can ignite quickly and consume the rest of the box. At the first sign of grease on the lower sides (within ~8" of the bottom, where temps will be considerably hotter than average) staple some steel flashing along that lower boundary as well. Then I think you can both cold smoke Thuringer and cook a brisket with minimal fire hazaard. As with any cooker, inspect after each cook.