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.