Hi Bee, Just a thought, having worked various events, your agency should be providing the bags and equipment. If for no other reason then liability, if your using anything more sophisticated then band aids, if something malfunctions that can be linked to a negative outcome, and it turns out to be your privately owned something, that puts everyone involved in civil jeopardy.
On the other hand, while I also carry a good sized kit in my car, I have no intentions of carrying it very far if I don't have to. You can do allot with a very small kit filled with well chosen components. My EDC is about the size of a grapefruit and at least remedially can manage anything from blisters to GSW/knife wounds. I think if I were you, I'd take my SAR kit and upgrade it as my knowledge and skills improved. But be selective when and where you use the personal kit. Anything mechanical can break. This is one of the reasons that we do rig checks prior to starting a shift. We need to assure that everything is there and functional. This happens twice a day in most systems. How many private folks do you know that check their stuff even once a month?

Regards Jim