Has donkey worn a bit except in lessons? If not, I would back up, put a broken mouth bit with ring sides on for eating meals with. Since you say donkey does not like a 2-part mouth, I would try a 3-part bit, with French link bean as the center part. Sometimes an equine with a low palate finds the multiple pieces easier to deal with.
I would just hang bit on a headstall, make sure the mouth is not low enough to clank on teeth. Might mean she needs a few wrinkles at the lip corners. Try for not tight, not loose, in fit. She has room to move it about in her mouth with her tongue. Let her wear it for eating grain and hay, an hour or two a day, over a week or more to learn how to manage the mouthpiece and food, not getting excited about being bitted, no chomping bit all the time.
Some fruit rollups on the bit will encourage easy bridling, press them tightly with your hand after wrapping mouthpiece to stay in place. Animal gets busy mouthing the bit, almost every animal LOVES the reward when you hold out the bit. Ours are like baby birds, mouth open, nose forward, for EASY bridling. Fruit-by-the-foot or Fruit Rollups on the mouthpiece are so much less messy than honey or molasses, lasts about 10 minutes of slurping and tongue play.
I would continue her driving stuff if you are doing training now. Goal is an animal who does not fuss with the bit in work or just hanging out, nothing about a bit and bridle to get excited about, unless it is food that comes with bridling in the stall!
When she is done with the eating and wearing bit time, I would work on harness bit fit in her mouth. We have very short lipped horses, shallow mouth they call it. You have to snug up the bits for work or the mouthpiece WILL be clanking them in the front teeth. Husband calls it “fitting for grins”, because they do have several wrinkles in the lip corners, “grinning” when bridled up. Just a different fit than other horses have.
For her tongue over issue, I would fit her bit just one hole looser than you prefer. This gives her room to work with, so she can put her tongue BACK UNDER if she doesn’t like the feel! I really advise against stopping to “fix the tongue” because smart animals QUICKLY realize you will stop if the tongue is flapping in the wind and do it A LOT! We had one horse who would flap his tongue when he wanted to rest. Took a number of “tongue fixing” before we caught on, and that was the LAST time we stopped for tongue hanging out. You know he only tried that once or twice again, then NEVER. You can’t stop to fix tongue in a class or competition, animal must DEAL WITH IT. Husband said “he got tongue over, he can get it back under by himself” and kept going. Horse fixed it pretty quick, no reward.
We have never found that tightening bit up high, to be of any help in keeping tongue under. Does make it harder for HORSE to fix his tongue problem himself though! So ignore tongue over while working animal, maybe fix it when you get ready to unharness with you on the ground, so animal does not think trick is rewarded. Maybe fix tongue after a few minutes of whoa and stand still practice, animal has already stood for a bit, not a reward or action/reaction thing to fussing or tongue flapping. Donkey is smart, they catch on FAST when you react wrongly.
and if you’ve only ever trained horses you can practically throw out the book and start over because they are different. My donkey will still throw his tongue over the bit if I haven’t driven him for awhile. I do stop and fix it and after a few times he gives up and keeps his tongue under the bit. I use a half cheek snaffle. It is the only bit I’ve ever used on him.