Simply operating an HID light on a circuit protected by a GFCI should not be a problem. However ...

Ballasts (and HID fixtures have ballasts) often fault to ground. That will trip the GFCI. The solution is to replace the ballast.

As for breakers marked "HID," here is what UL has to say: "A circuit breaker rated 50 A maximum, 480 V or less, and intended to switch high
intensity discharge (HID) lighting loads on a regular basis is marked “HID.” "