It should be possible to construct scenarios where a logical register is mapped to "physical register + immediate" where the immediate depends on the path taken through the program so far. The easiest example would be an if-statement that optionally adds 1023 to a register, and then an add of 1 after the if-statement. The add of 1 overflows the immediate stored in the register renamer if and only if the if-condition was true.
As you said, that requires taking different control flow paths through the program, which is something to be avoided by constant-time algorithms anyway.
One potential twist: This shows that it may be important that the interface between the constant-time algorithm and the rest of the program goes entirely through memory and not through registers. If the interface does go through registers, it's possible that variable register-renamer-immediates leak from the outside world into the constant-time algorithm. (This admittedly feels like a bit of a stretch, but who knows.)