First off, the GFCI Breaker is a (literal) snap to replace. You remove two wires, pop the breaker out, pop the replacement into its place, replace the two wires and you are done. Now, if there is a MAIN BREAKER in the panel and you don't mind resetting all the clocks in your house, flip it off first - a safety measure that I have and have not used when doing this operation many times over the years. I am the guy who regularly replaces switches, outlets and fixtures while the subject circuit is hot - you may not be so comfortable first time out.
You can replace the GFCI Breaker with a normal breaker of the same Amperage rating - just connect the BLACK lead for this test and ignore the other. If this breaker accepts the load, then either purchase the expensive GFCI replacement or replace the first outlet in the circuit with a GFCI Outlet and run the other two from it (to be protected)'
A GFCI device 'detects' a 'Ground Fault.' That is, a connection as between the HOT and any Ground - most especially a Human Ground and is designed to be relatively sensitive to such an event (so as not to shock a human to distress or worse.
As such, a 'DAMPNESS' in one of the exterior fixtures (or in any electrical box on the circuit) could cause the GFCI to trip and it is conceivable that it might not trip if there was no load on the circuit. I have NOT experienced this situation.
Thus, my first thought was that the Exterior lamp fixtures might be the culprit. Then, you wrote "I have opened up what I believe is the first outlet in the circuit, " However you did NOT confirm your supposition - that is, you did not indicate that "unhook(ing) the outgoing connections to disable the rest of the circuit" actually worked - did you verify that the overhead light, remaining outlets and the exterior fixtures were 'powerless' after this?
Without an answer to this question, it is difficult to provide a solution.