How to discover LCD controller model
It's hard to tell the controller which is been used by the chip. Many random LCD won't have some valid LCD part number where you can google and search it. Only thing you can do is try to reverse engineer the LCD, as you found its SPI and program a microcontroller with the same SPI data and check for it's correctness. If that works then, then dig deeper with the LCD with different patterns and program it. 

The only way to know to drive these kind of random LCD is reverse engineering with MSO ,collect the data and analyse the data and check for it's correctness using a microcontroller ,interface with that LCD and check with the data that you analysed. It takes more time to reverse engineer these kinds of LCD but still it's possible do it. I also reverse engineed an random LCD which came with phone(dotted LCD) . It really took very long time to understand the protocol, it's function and still there is literally more things that I have to explore with that LCD. It doesn't end at all.