A CC LED driver doesn't have a definite voltage output. The voltage will be determined by V(f) of connected LEDs, in accord with their (individual) I-V curves.
If you LED has V(f) = 1.6 V (+- 0.1 I guess, depending on current and temperature), you can connect 3 LEDs in series to your CC driver. Then the total voltage would be 4.8 V (+-). This should be enough to power the Raspberry Zero.
As a matter of fact, LEDs can be used as "parametric voltage stabilizers" instead of Zeners, with about same quality of voltage regulation. So 3 LEDs should form a sufficient stabilizer, since RPi uses PMIC (PAM2306AYPKE), which is tolerant to 5.5V. So this should work if you put 3 LEDs in series.
The only small concern is that the Rpi might have variable power consumption, which will eat into your fixed CC budget, and steal some current from LEDs as Spehro Pefhany noted. It will have some effect on LED brightness.
If constant brightness of LEDs is important, you will need to use a separate DC-DC 12V-5V converter to feed your Rpi. Or make your Rpi code to consume constant power, and adjust the CC limit accordingly.
Of course if one of your LEDs suddenly dies (which do happen with them), your Rpi might be in trouble, so you might want to design some extra protection from overvoltage unless your CC doesn't have one. But you said you can set the driver to no more than 5 V, then you should be fine from all angles.