I feel like I'm very close. To start out, I found an alternative FBCP driver for the hardware combination I'm attempting to get working available at https://github.com/juj/fbcp-ili9341. I went through the initial build procedure and it seems to be working with a few exceptions:

After looking in the documentation, it explicitly says touchscreen will not work, but one may be able to get it working if they took a stab at it. After that, I switched back to the Adafruit FBCP. I found that if I run FBCP at the command line after booting it, it seems to work just fine with touch and everything. I tried placing it in the /etc/rc.local file, and it worked for a second, but then another copy started up and it stopped working. After running grep to search for fbcp, I found a service used to start it up. When I type sudo systemctl restart fbcp, it starts right up. I looked to see if anybody else was having the same issue and I found an open issue here which describes my symptoms perfectly. Just after posting a comment on it, it was assigned to somebody to fix, so I'm hopeful that all I need to do is wait for that.

Now as for the keyboard., I found an article that describes how this can be accomplished with setting up a GPIO Matrix Device Tree Overlay and a couple other components. I also came across a couple things that briefly mentioned using diodes to make it so key presses can't be mis-detected. I'll have to do some research on that.

I started mapping out the circuit board for the keypad by taking a picture and importing it into photoshop, but unfortunately I ran into an issue. The circuit board from the TI-86 is so corroded that some of the traces are completely dissolved. That's on top of the complexities of it that it isn't in a perfect matrix. At this point I have the option of repairing the board which would mean some ugly Frankenstein wires and probably having reliability issues down the road or sacrificing the TI-83 that originally came out of the case...with the screen lines that take 10 seconds to reappear...and the now missing battery panel...yeah, I think I'll be sacrificing the TI-83. I know that board is in pristine condition.