With no rear brakes at all and no ABS warning, I would be looking for a big air bubble in the system. The FJR rear brake is linked to one of the front piston pairs (lower right front). When bleeding brakes, you have to bleed the lower right front piston pair using the lower bleeder on the right caliper (and using the rear pedal - fluid comes from the rear reservoir). You then bleed the rear caliper. While you're at it, remove, clean and lubricate the rear brake pedal pivot - should be done annually.
For the front, I usually bleed the left before the right, but it doesn't really matter much. It is a good idea to activate the ABS pump before and after bleeding. It may be done using the following procedure.
https://www.fjrforum.com/threads/test-and-maintain-your-abs-system.136000/#post-834610
Do your clutch fluid at the same time. Service both hand levers (clean and lube). Make sure you remove, clean and lube the brass bushing in the clutch lever (the one that engaged the clutch push rod). If not maintained, it will wear out and become ineffective - properly maintained, it will last the life of the bike.
Clean and lube shifter linkage too.
Note: I hope you got an '06. '07 ABS pump. They changed for '08+.
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