California Vehicle Code Section 21453 (b) regulates right turns on red:
Except when a sign is in place prohibiting a turn, a driver, after stopping as required by subdivision (a), facing a steady circular red signal, may turn right, or turn left from a one-way street onto a one-way street. A driver making that turn shall yield the right-of-way to pedestrians lawfully within an adjacent crosswalk and to any vehicle that has approached or is approaching so closely as to constitute an immediate hazard to the driver, and shall continue to yield the right-of-way to that vehicle until the driver can proceed with reasonable safety.
No exception is made for intersections where bicycles can go straight, so I conclude that right on red is legal in that case as well.
Logically: it is legal to turn right on red from a lane where all traffic must turn right, and it is also legal from a non-turn-only lane where all traffic is allowed to either turn right or go straight. There is no reason for this intermediate case, where some vehicles may go straight and others may not, to be different from the two extremes. Especially since, when the light is red, neither bicycles nor cars nor anything else should be going straight through the intersection.