I am OK with bananas, peeled apples and pears (Asian pears are a yummy variety to mix things up), but I've also done well with non-tart tropical fruits like mangos and papayas (if you can't find fresh, you can buy in frozen chunks). Nectarines, peaches and cherries too, although I avoid imported ones this time of year...wait until summer. And don't forget avocados, which are great for you and well tolerated. You might try cooked ripe plantains too (peel turning yellow and black): cut a slit through the skin lengthwise and microwave; or cut into quarters with peel on and steam, then mash; or peel and bake; or peel, slice and pan fry.
I'm undecided on the canteloupe and honeydew melons.
Fruits I avoid when flaring:
Grapes, especially with the peel on (I can tolerate a few peeled grapes)
Citrus fruits (a little lemon or lime juice in food is usually OK)
Strawberries
Large helping of blueberries (too much skin I think)
Blackberries (too many seeds)
Kiwi
Pineapple
Plums
Watermelon (but I love it too much, so I cheat, and I see it on the other end

)