I had never heard of the "big Pro Mini". I am suprised tha they did not think of a unique name.
Your 2.4" shield should work in both Uno and "real" Mega2560. I suggest that you try again. Preferably with a new Mega2560.
I have only tried the Teensy3.2 with a few controllers. It is too fast for the ST7789. ILI9341, ILI9325 are fine.
No, I do not have a Teensy3.5 or 3.6. It would be trivial to support them. The GPIO speed would definitely need to be slowed down.
I do not see your 3.6" problem. The pcb will accept the 3.5" shield. Or any size of shield. If you have an enclosure that only fits the narrow 240x400 shape, you need to find a 3.2" 320x480 panel.
Incidentally, if you are using a custom pcb, you might just as well map the data bus to PORTA on your mega2560 chip. It gives a much faster display.
David.