Possibly, and yeah rebar might be overkill for this. I was thinking more like a steel mesh near the bottom, or fiberglass. You can get cut fiberglass for concrete fill like this:

https://www.amazon.com/Fiberglass-Concrete-AR-Glass-Chopped-Reinforcement/dp/B0FG2JGN48

This makes the concrete stronger and lighter and increases temperature stability, and it's pretty cheap but it does add some complexity to the casting. I don't recommend handling glass fiber with your bare hands - glass splinters suck - and you should also wear a filter mask (hopefully you're wearing one while mixing the concrete anyway because silicosis also sucks). I found this video which shows the process for making a countertop with glass reinforced cement:

https://www.youtube.com/watch?v=pYqVPdbi6mM

They use an epoxy-based Portland cement - you would not want to do that because the epoxy might outgas when heated by charcoal, and you wouldn't want to do the urethane finish coat for the same reason, but otherwise you should be able to follow the same process with water-based concrete.

I haven't actually tried this myself, but I think having the fiber mixed in would be especially good for keeping the concrete stable over repeated cycles of heating/cooling expansion and contraction.

All that said, I think you should temperature test the piece you've already made first before investing time/money in doing something more complicated. If it works then it works.

Nope.