It is probably possible, but SPI on the MT7688 has enough quirks, along with the current state of the software stack, that it may be quite a challenge to tackle this in a pioneering way, unless someone has already gotten your display or a conceptually simpler one working.
An idea you could consider would be using an Arduino (or bare ATmega or some even cheaper MCU) to act as a serial-to-SPI bridge, such that you could just write data out the serial port and the Arduino would turn that into commands for the display. Wasteful of hardware perhaps, but if it transforms the project into a more familiar one it could be beneficial. Note that in doing it that way you also abstract out the Omega entirely - you can probably run your program on your development machine itself displaying through the Arduino while you work out the details, and only bother with cross compiling and installing it on the Omega module occasionally as a check. You also end up able to replace the Omega module with whatever platform seems interesting next year.