How hard is it to reverse-engineer a two-way protocol for a beginner with a logic analyzer?
I would start by looking at the PCB for the driver chips that are used (RS232? RS485? TTL? analog?). This gives a first clue how to look at the signals.

It won't hurt to have some idea of how an asynch serial, I2C, SPI, etc signal might look like. Google is (of course) your friend.

You could try to vary the inputs (speed, power, etc) to see what changes in the signals.

From a designer point of view, I would imagine how I would build such a system (or how others have done it) and check whether it's done the same way here. I would probably use a dynamo-like thing in the wheel-blob and do all processing in the display unit. (Which would make it very hard for you to interpret the signals).