Why does my GFCI circuit breaker trip with any small load, even after replacing the breaker?
Since it's a Ground Fault Circuit Interrupter (GFCI) breaker, it's very likely that you have a ground fault.

GFCIs work by detecting an imbalance in current between the ungrounded (hot), and grounded (neutral) conductors.  This is why you only see the trip when the circuit is under load. If there is 0 current flow, the current is balanced.

I'd start by inspecting the receptacle.  Turn off the breaker, then open up the box where the receptacle is installed.  Look for damage to the receptacle itself, or water/moisture in the box.

For outdoor installations you should always have a weatherproof box, and a cover that is weatherproof even when the receptacle is in use.