New Shore Power Pedestals sensitivity

Down and dirty.
Turn everything off on the boat.
Plug in shorepower.
Sequentially turn loads on one at a time until it trips your shorepower.
Now you've found the circuit that is causing the issue.
Should I assume that the inverter is Marine rated and installed by a professional?

If you have a Marine inverter it will only bond N to G when in inverter mode. If your inverter is not marine, the N and G will be permanently tied together and it will trip the new pedestal with GFI

If your inverter ties green and white wires (ground and neutral) as in a house, It will trip the GFCI breaker in the dock power pole.
Also tied green and white on the gen-set will cause this.
Un-like a house, where green and white are tied at the breaker box, on a boat it is tied at the dock power pole and must not connect on the boat,,,,, until off the dock. Then & only then are fault and neutral are tied. A proper transfer switch provides this for the gen-set.
A relay provides this in the marine inverter.

Then some way, some how, somebody tied green & white together or backwards.

actually not always. Some appliances like ovens, stoves, fridges, washers may tie G and N inside the appliance. When you turn off the single pole breaker, you turn off the most Hot not the Neutral which remains tied to G

If your north of Boomer, I'll be down there soon.

Double and triple pole breakers are the most help on shutting the lines down, That way, you know the white wire is turned off. Single pole breakers just turn off the hot wire.

There was a customer that had white and green tied in a clothe dryer for house hold use. Rite on the back was the wire diagram and notes to remove that strap when needed, like on a boat or RV. I had to run a white wire to that 230 device, the dryer timer runs in 115Vac.

On our own Bertram we had an intermittent bug. Every time we visited a new dock with these fancy new dock breakers, plug in, turn on the A/Cs, no trip all, was happy. Later when cooking dinner, sometimes that breaker would trip. All always worked well until you plugged something into the 115Vac outlet on the oven face. We started to realize the oven & stove lights on at the same time may have something to do with our GFCI problem also. Finally found the lights and outlet's white wires were tied to the green for return current. Figured Bertram got cheap one day and did not hook up the white wire to our 230Vac oven/stove.
Once that was fixed, our 42 year old Bertram has never had a problem with one of those new power poles.

I did get a laugh on that last line; inverter is Marine rated and installed by a professional.
Rare and far between; a professional and correct inverter install.

And that’s why I prefer anchoring out... no worry about power !! Even here in the Exumas whenever we dock at Staniel I end up juggling loads and usually during the day end up running half the boat on shore and half the boat on genset. The 100amp is never quite a 100amp. Couple of day ago I started the convection oven and could hear right away it didn’t sound right. Sure enough... 204 volts.

Another reason, we usually do not touch docks unless we really have to, like an extra 200 gallons diesel at Marsh or Spanish, to make it further up the east coast on our way home.
We don't charter. Just family at times. In/out was usually Treasure or Marsh on a dink or ferry.
Inverter and gen-sets do us very well. Sadly, long over due to check on our friends face to face. So far, all are good.
When the new docks are in, not sure if they will be GFCI since they really do not want to chase anybody away, like state side docks..

Yes we are.
Been burned to many times at Boat Harbor. I could sense they did not want to serve us. Frak them.
The kids at Spanish and original Green Turtle resort luv us.
Bless them.
We are anti social and they, our friends and better vendors, understand.

Yes, it's been 4,5 & 6 years trying to stay at Boat Harbor. Fuel dock is fair, unprotected from SE wind and a bear to tie to.
Restaurant sux and resentment you were there.
Always tried to schedule BBQ night at the Jib Room. Been hitting there thru 4 owners, near 40 years.
Yes, we visit our friends, understand quickly who is not.
We tried that slut bar over from Taloo Cut once years ago, Never again.
Yes, we pick our landings, very few, but know where are friends are when we do land.
We always hit Pete's Pub. Always a new face. Always treated well. Just wish other Abaco shops followed suite.
Now that I think of Elbow Kay, Frak the Frakers in Hope Town.
Noses pointing to far upward to talk to.

Yes we anchor out, keeps the tension down.