If you use the art in the Star League sourcebook as a guide, you can go with a full-body coolant suit, which effectively takes care of any/all modesty and climate-control issues. (Essentially start with a jumpsuit and stitch rubber tubing all over the outside).
This would be useful, and you can even cheat a bit. Make the rubber tubing actually carry water, and if the convention gets too hot, you go back to the room, and put an ice pack in a storage section. Not just a costume, but an actual working suit. You can look for medical diagrams showing where emergency rooms put ice packs when treating heat victims, fur locations where to put more cooling tubes.
Or to avoid the potential for leaks, just put copper or aluminum wire underneath the coolant vest, but in the same locations, and have the ice pack touching the central location. The metal will help conduct the coolth around the body.
From there, look up information about what modern tankers consider part of their gear when they have to bail out. First Aid kit, gun, survival kit, similar gear. Make it practical, rather than fanservice.
Plus you need an outline of a Mech on it so people can recognize it as BT cosplay, not any other group.