For what it's worth, it's vanishingly unlikely that *any* of the cables in the
box containing the GFCI and dimmer has its other end in the box containing the
4-way switch. When controlling a light from 'n' locations, you need two 3-way
switches and (n - 2) 4-way switches; e.g. for 2 locations, two 3-ways and zero
4-ways; for 3 locations, two 3-ways and one 4-way; etc. -- and the 4-ways
*always* sit in the middle, with the 3-ways at the ends. Power goes to one
3-way, and the light is connected to the other. Since the power for the island
lights is clearly routed through the GFCI, there must be a cable leaving that
box and going to one of the 3-way switches -- *not* the 4-way.
Well, I'm no so sure about that. The island lights are on a different
circuit than the dimmer/ recessed lights. I have to turn off two
separate breakers to disconnect both. I assume they share a common
neutral. (yes, this is where I'm glad I have a voltmeter - there is no
way to tell whether the power is off for the island light switch
otherwise)

Sorry, I guess I wasn't clear -- your diagram does not appear to represent an
electrical circuit that can actually function. Maybe I'm misunderstanding or
misinterpreting it. Or maybe there's a mistake in it -- have you
double-checked it to be sure it accurately represents the reality of what's in
the box? (Referring specifically to the diagram of the GFCI & dimmer wiring)
I've checked it twice, but like I said, there is a LOT crammed into the
box that houses the dimmer and GFCI. I've lost daylight (chicken and
egg problem....) but will re-re-recheck tomorrow.

Yes, the recessed lights turn on regardless of whether the island
lights are on or off.
Interesting. And somewhat surprising.
What about the recessed can lights? Do they also go out when the GFCI trips?
No, the recessed lights stay on.

Does anything *else* go off when the GFCI trips? Any outlets that go dead?
Yes, there is an outlet on the other side of the kitchen that goes
dead. One of the three switches that controls the island lights is
above that outlet.

(I'm betting on 'yes' answers to all of the above)
Almost. That is what I found so puzzling about my problem originally
(GFCI tripping when island lights were switched on). They're on
separate circuits, the only common element is the neutral.

Some more info.
1. When the island light switch is OFF, the voltage for both input
("black left" in second diagram) and output ("black right" in second
diagram) is ~80V. It doesn't matter whether the dimmer is on or off,
voltage is the same. I believe this is normal for a 4-way switch when
you measure using a voltmeter
2. When the island switch is ON, the voltage is 0V for both input and
output. Again, doesn't matter whether the dimmer is on or off.
3. If any of the other island light switches are switched on, the
voltage drops to 0V as well.
4. When the dimmer is OFF, the voltage between the two black wires
attached to the dimmer switch is 120V. When on, the voltage varies
between 12V (maximum light output from recessed lights) to 100V (almost
no light output from recessed lights)
5. The GCI outlets supply 120V, regardless of any switch position.

Not to confuse the issue, but here's another piece of information.
There is a second set of recessed lights in the kitchen. The second set
is also controlled by a dimmer (in an outlet box that also has a GFCI).
The first dimmer is located closer to the island switch, so I didn't
notice this before, but....if I turn on the second dimmer, then the
island lights work, even with the first dimmer off. In other words, to
turn on the island lights, I have to turn on the island light switch
and EITHER dimmer.

Shot in the dark (no pun intended): There are a LOT of neutrals tied
together. I can't easily get to it, it's wedged tightly in the back of
the box. If there is a neutral loose, could that explain the symptoms?

Thanks again.