Shimano MT500 leaking from bleed port
It could be an easy fix, or a new brake. Are you prepared to trash your travel plans over the cost of new brakes?  Given the time frame, I would either hand it over to my LBS, or put on a new brake and sort it out when I got home. Time frame for sourcing a new brake means any hiccups and you won't have time to sort it unless you buy from your LBS. Your best option is probably get the LBS to do the work as well.    

If you replace the caliper on its own you will still need to bleed the brakes. Replacing the entire brake assembly, if you get one with the correct length hose, should require no bleeding. Shimano (Don't know abut others) ship with the hose detached from the lever so you can install into the frame if needed. Installing a new brake is much simpler than bleeding, especially if you run the hose on the outside of the frame. 

Only thing you really need to decide is what quality brake to buy. Your could go cheap - say MT365, with the plan to fix up the MT500's later, or use this as an opportunity to upgrade to M596 or better. (Going to M596's you should also upgrade the disc rotor to the narrow shimano version - MT500's use the wider Shimano rotor)

I recently needed a new back brake in a hurry and got discounted MT365's with a view that the $US20 cost could be saved by waiting for a discont on XT's. I still have not upgraded, the 365's are not great brakes, but for the rear give me enough power an modulation for my style riding.