This is why it's illegal for you to do what you're doing.

If the GFCI trips with a load, it is almost assuredly because one side of the part of the circuit with that load is connected to the protected side of the GFCI while the other is not.

I have this strange feeling you've connected the neutral to the load side of the GFCI but booted the hot from somewhere else. What switch?

Depending on where you are and when the house was built, all the countertop outlets should be on a GFCI. I suspect that since the GFCI is also feeding things that are outside the kitchen this is an older house OR some other hack has already introduced code violations.

from the exterior receptacle>> The black wire goes to a switch and then to the LOAD side HOT terminal of the GFI
The white wire goes to the LOAD side NEUTRAL terminal of the GFI...is there any reason this setup should not work?

Even if you can get it to work, you are not allowed to feed an outside receptacle from the kitchen counter small appliance branch circuits.

Well, there's also the possibility that there's a ground fault somewhere else.

I would abstain from providing advice for an illegal installation.

This sounds like one of those situations where a code is written just to pump more money into the hands of electricians..to tear apart a house from one side to the other to install a dedicated circuit for an exterior receptacle is ridiculous..Unless you can explain to me the safety issue with it.

"This sounds like one of those situations where a code is written just to pump more money into the hands of electricians..to tear apart a house from one side to the other to install a dedicated circuit for an exterior receptacle is ridiculous..Unless you can explain to me the safety issue with it."

You do not need "to tear apart a house from one side to the other to install a dedicated circuit" if you now what you are doing.

The safety issue is that the kitchen counter circuits are 20 amp small appliance branch circuits, are sharing them with other loads can result in overloads.

The 'safety issue' is also that the NEC when adopted by jurisdictions normally has the force of law.

Luckily your homeowner's insurance does cover your own mistakes.

Unluckily his homeowner's insurance won't cover the mistakes he's making to his friend's house he's "helping."

You are very welcome for the free, experienced advice. Have a great day.

Signed,
Your local electrician.

"This sounds like one of those situations where a code is written just to pump more money into the hands of electricians..to tear apart a house from one side to the other to install a dedicated circuit for an exterior receptacle is ridiculous."

What an absolutely moronic statement.

"This sounds like one of those situations where a code is written just to pump more money into the hands of electricians..to tear apart a house from one side to the other to install a dedicated circuit for an exterior receptacle is ridiculous."

true. you cannot be doing something outside and doing something inside at the same time. at least it is not as bad as what the last people did to my house. they ran 12/2g to my back porch light from the counter recs. then, they ran 14/2g for a loop switch. i had to terminate the 12/2 in the light box and then use the 14/2g to powerfeed the light. the wire in the switch box was also 14awg on a 20a cb, so i had to downgrade the breaker size in my terrible ge panel.

Smitty, Please do not espouse this illegal and unsafe nonsense. Even after your alleged "downgrade" you're still got an illegal installation.

As far as the OP, you're allowed to DIY your self and your family to death on a single family home but that doesn't extend to others. The truth of the matter is the GFCI may have been indeed grandfathered as it was previously hooked up and it was probably hooked up right, but somehow in your screwing around, you've introduced a ground fault that now needs repairing and we can't do it over the internet. You'll have to have your friend suck it up and call an electrician to fix your mistake.

Serious? NEVER wire kitchen circuit to anything else.....it consumes enough via the toster oven and coffee maker. (1) What you are doing is WRONG, before you even address the tripping problem.
(2) NEVER backstab. Use the screws. Screws already used?...then it's time to pigtail. Not enough room in the box...yet, another reason not to be doing what you are doing.

In Ithaca, they would shut down your friends electrical service until you get reinspected, if you mention what you have done, and the failure to get a gfi, and a simple switch to work together. SHUT DOWN...for about two days....