GFCI trips when lights go on
Nope, you can't go in a loop like that...

You have power going in a big circle.  Power flows from "J-box", to the GFCI device, off its load to the outlet, then through both 3-way switches back to "J-box" and on to the light.  Neutral does not flow back the reverse route.  That's a big big problem, because the currents in each cable are not equal, which means you're inducing a lot of EMFs, getting eddy current heating, and other badnesses.  You need to fix that, separately from the GFCI problem.  

Generally, non-UK wiring is done in a physical "tree" topology.  Wiring may zig up and back several branches, but if you stuck a clamp ammeter around all the wires in a cable together, it will always say 0 since they cancel each other out with equal flows.  Yours does not do that.  

There should be /3 cable between the two 3-way switches.

The GFCI is a separate issue.  You must always power things symmetrically, but with a GFCI it's actually going to cause a trip. In this case the light gets its "hot" off the GFCI LOAD terminals, but gets its neutral direct from source.  That means the hot current leaving the GFCI is not matched by an equal amount of neutral current returning, exactly what they are made to detect.   

Smart switch option 1: radio/wireless/powerline coding

This works for smart switches which do not need a communication wire. 

You will need a "master" smart switch and matching remote, or at least, units designed to work in the same home-automation system.  The remote must only take always-hot and neutral, and relies on radio or powerline comms to talk to the master.  

Fit the "master" in the right 3-way position, connecting its always-hot, neutral and switched-hot to the wireds readily available here.  

On the left 3-way, the wire heading north from COM, remove it.  Boom, it's gone.  Loop broken.   

The two wires between the 3-ways are redesignated for always-hot and neutral and carry power to the remote switch.   If the remote is battery operated, you don't need this pair at all.  

Smart switch option 2: communication wire

Again a master and remote designed to work with each other.  But the search for a suitable unit will be harder. 

First, on these, you must consult with the manufacturer and ask them if the switches can be on two different services.   Say it just like that,  not subpanel, different service.   If they want more, tell them they will be on opposite sides of a GFCI.  And get the manufacturer's advice.  To be clear, when I say "manufacturer" I do not mean the former landscaper former meth-head who now works in the paint department at Lowes and you caught coming back from the bathroom.  Best advice: don't buy anything like this at a big box, buy from a local electrical distributor.  Best advice: don't buy anything from a big-box.  

In this case, you hook up the master in the right 3-way position as before; grabbing available hot, neutral and switched-hot.  

On the left switch, you are already bringing down always-hot from the passthrough... Grab neutral there as well.  Note that this switch will be powered from inside the GFCI protected zone.  That should make your spider-sense tingle, hence, the conversation with the manufacturer.    Then you use one or both of the messengers to satisfy the requirement for the communication wire(s).