Should my receptacle tester be able to trip a GFCI breaker in the panel box?

If the receptacle is grounded: Yes it should trip it.

If the receptacle has some semi-grounding that is insufficient to clear a dead-short bolted fault (i.e. incapable of flowing 200A without setting the house on fire): Yes it should trip it. And that is fine. A bolted fault is also a ground fault, which should trip the GFCI, ending the event.

If the receptacle is not grounded: No, it should not trip it. That would be disturbing if it did.

If the receptacle is not grounded, but a tester indicates grounded: Then a) no, it should not trip it, and b) that is a bootleg ground that defeats the entire purpose of even having a GFCI /facepalm /Darwin_Award and needs to be corrected ASAP.

If you stick the GFCI tester in a 2-prong cheater and come off the ground tab with a separate wire run across the house to a part of the electrical grounding system: then yes, it should trip it.

Harper's answer is, as usual, totally correct. But to explain why:

GFCI (whether breaker or receptacle) detects a difference between current on hot and current on neutral (or two hots if 240V, but we can ignore that for simplicity).

The internal test of a GFCI does whatever is necessary to test.

An external test - e.g., built in to a 3-light "magic 8-ball" tester - has to cheat a little bit. It takes a bit of the current that would normally be going from hot to neutral and sends it to ground instead. If the ground pin is properly connected then this works and the GFCI trips. If the ground pin is not connected (which can be legitimate it properly labeled when using a GFCI) then the external test tries to send current to ground, but it has no place to go and so that circuit is not completed and the GFCI does not trip, but this does not have any effect on proper operation of the GFCI.

The one exception, as noted by Harper, is a bootleg ground. This is ground connected to neutral in the receptacle. If you have that, a 3-light tester will show everything "OK" (because it can't tell the difference between neutral and ground) but the GFCI test function of that tester will not work. That is the one combination which is an indication of a serious problem. "Open ground" + "GFCI test fail" is OK.