If a GFCI tester trips a circuit breaker is there a problem
GFCI tester test + GFCI = tripped GFCI

A CHGF is a GFCI and circuit breaker in the same package, using the same breaker contacts and trip mechanism.  So, when your inspector pushed the test button on his receptacle tester, shunting a bit of current to ground, the GFCI-breaker saw the difference in current between hot and neutral generated by this and tripped, just like it was designed to.

This also means you don't need to connect a GFCI receptacle to that circuit -- it's silly and redundant at that point, and also leads to the coordination problems you're seeing where two GFCIs are tripping in reaction to the exact same ground fault.  Thus, the easiest way to solve this is to replace the GFCI receptacle on that circuit with a regular receptacle with a "GFCI Protected" label on its front and let the GFCI breaker do its job.  

(P.S. a more reliable test of a GFCI is to use the "test" button on the unit -- that shunts current from load-side-hot to line-side-neutral, which will also trip the GFCI, but does it independent of any problems with the equipment ground to the outlet.)