Update:

When I add the landscape light to the Load terminal in the GFCI combo device
(instead of a separate switch), it works. Now the GFCI works for both the
pool light and landscape lights. Except now I don't have independent
control over them. That's fine because the landscape lights are next to the
pool anyway. So this way I have GFCI protection for ALL of the outdoor
lights near the pool.

Why does this GFCI tripping happen with a separate landscape light switch?
My guess is that by turning on the landscape switch, it causes the current
to drop unevenly in the circuit legs and the GFCI senses a slight mismatch.
So by combining them together the current drop is evenly matched.