"You could take the horse and burro to the show."
I would be very careful with this suggestion. Some horses have never seen (more importantly, heard) a burro. It could cause a commotion if you take the mare away to show and the burro freaks out or gets loose somehow at the showgrounds. Young kids or beginners at a first time show may get hurt.
If you can trailer her to a show ans just cruise around the grounds with her, that would be ideal. No pressure for you, no pressure on her. Goal is to get there, walk around, and stay there until she settles in.
Just take your time with this horse and set realistic expectations for both of you. Desensitizing her will not happen overnight

and it helps to get a good game plan together.
I have found the following very useful to get extremely nervous or buddy sour horses used to moving outside their comfort zone:
Tack up, then some longing to get out the "friskies". Not too much, just some walk, trot, and canter in both directions. You just want to warm them up a bit and pay attention, not wear them out.
Undo the longe line and put a halter/lead over the bridle. Lead from the ground and go for a walk outside the property (as far as you can without a war). Talk, sing, hum, whatever, while you walk. Vary the route or the length of time every day, if you can. If the horse *freaks*, redirect their attention to anything other then negative behavior: let them munch some grass for a second or head them down a different direction - just not towards home! Don't let the horse make that decision for you.
Sure the neighbors will think you're nuts, tell them you both need the exercise and you're training for a powerwalk marathon.

Come back to the farm. Longe for a few minutes more, just walk and trot.
Mount and do some walk, trot and canter for a bit. Then head out for a quick walk back to the end of the street, mailbox, property line... wherever you are most comfortable and she keeps behaving.
Then untack, groom, lots of love. Carrots as earned

Before long they will associate leaving as not a bad thing and to do it well means rest. Freaking out on the road doesn't mean no work when they get home. They pick that up fast.
Soon you can drop the second longe, or the second walk, or go out for longer periods of time. Incorporate some work on movements while you are on your outing. Use what works for you, but just don't lose your cool - it may take a day or a month for her to see that outside isn't the end of the world. Repetition is key!
Act as if you have no time constraints, because most likely you don't. She sounds like she'll be around for a while.

Hope this helps,
Marianne
ps: maybe it would help to contain the burro in the barn or distract him with some hay/treats while you work with her. This may not be possible, but if it is, it may help you.