Refrigerator trips new AFCI/GFCI circuit breakers
Kitchen counter receptacles need to be GFCI protected.  Refrigerators do not, and should not.  

A refrigerator is simply not what GFCIs are for, like manassehkatz says.  All the electrical bits are in the bottom back, and you're not likely to knock a refrigerator into the sink! 

The fridge circuit definitely does not require GFCI, however if the fridge is on a shared circuit with other outlets which need GFCI, then that does not give you a right to remove GFCI from those outlets!  Your inspector is teling you the fridge needs GFCI because it's on a shared circuit with countertop receptacles, and those need GFCI.  The fridge does not unless there is a local amendment to code, but it would be an awful amendment! 

So the fridge should be removed from the GFCI, either by 


running a dedicated circuit with one plug, 15A is allowable; or by 
changing the breaker to non-GFCI and using GFCI deadfronts or receptacles in appropriate place(s) to protect the kitchen countertops, as Code requires. 


The inspector has no "right" in the electrical code to compel you to use GFCI breakers particularly, though they do provide better protection.   

The fridge circuit may require AFCI, but that's a different thing than GFCI and the refrigerator is less likely to have a problem with it. 

The ideal setup

The ideal setup is a dedicated circuit for just the fridge, with a one-eye receptacle (not the usual twin) behind the fridge where nothing else can reach it, a breaker that exactly matches the receptacle amps (a requirement on 1-socket circuits), and a labelmaker sticker on the receptacle saying "Refrigerator only". 

Even more ideal, I would also run it in EMT conduit and metal boxes, and make the case to the AHJ that the AFCI protection is not needed due to the EMT. 

The nuclear option

If it is infeasible to remove the refrigerator from GFCI protection, you can remove the GFCI protection from the refrigerator! Use a Kill-a-Watt to measure how many VA the fridge uses when running (not starting), double it, and obtain an isolation transformer of that many VA.  

All the GFCI will see is the primary winding of the transformer. What happens past the secondary winding, stays in Vegas. Ground is passed straight through the isolation transformer, so the fridge is still safely grounded.  If the fridge has a hot-ground fault, that will do nothing at all.

Since the isolation transformer is a plug-in consumer appliance like a power strip, it is out of the inspector's jurisdiction. He can't say a thing about it. 

Alternately you could use a component isolation transformer and build it into the refrigerator.  Altering an appliance is not an Electrical Code matter, so again, outside his jurisdiction.