[1.16.5] Associating a TileEntity with an existing vanilla Container?

[ninety]

Apologies in advance if I'm going to be missing anything fairly obvious, but I'm not quite sure what approach to take here.

I am attempting to implement an automated version of the vanilla Smithing Table such that it has its own tile entity and item handler, and allows items to be piped into the input slots and extracted from the output slot. Hence, I figured making the container for this table extend SmithingTableContainer would be enough to start with, though now I'm not sure how I would go about tacking the desired tile entity behaviour onto it and I'm very likely cutting a few corners trying to go about it this way. Is this approach feasible in the first place or would I be better off extending the pure Container class and re-implementing most aspects of SmithingTableContainer in this new class?

Should go without saying that I am very much a novice with Forge still; the last I used it (circa MC 1.8) I had next to no idea what I was doing from a programming standpoint either.