Replying to myself here, but it started working! I can now see live data and read/clear DTC on my GSX-R. The refresh rate is poor (~1Hz), at least on racechrono, probably since the baud on BT is 9600. I tried changing it with BT BAUD8, to no avail. I will need to try the UART way, maybe that will be successful.
Even though the baud rate is causing me issues, I am confident enough now on the HW and SW side to move from the breadboard to a final PCB. Thank you @TriB for the support.