Your bari is a fairly common and easy to find horn....you really should look into borrowing while here or possibly getting your group to secure you a rental.
Depending on the airline and the mood of the stews, you
can get baris on board in a gigbag. I have even managed to get mine onboard in a BAM case.
That being said, if you have to put the horn in the cargo hold I suggest this:
Cork down all your keys, then put LOTS of bubble plastic around the bell, bow and curl and under the body tube anywhere there is no padding support in the case. Fully (top to bottom) secure the horn with bubble plastic, so that there is no way it can move even a millimeter inside the case
Then wrap the case in a good layer of bubble wrap, and put the case in a cardboard box with at least a couple of inches of packing peanuts all around. Tape it up and take it with you as air cargo.
Put lots of FRAGILE INSTRUMENT stickers on the box, and pray.