A nice start might be something like the "Comprehensive" kit from AMK. That's what we currently take with us on troop trips, augmented with some extra stuff. The key thing is to have people properly trained, and to regularly inventory the kit and restock it. I think the retail price for the kit is under $200.
They have a couple smaller kits that may fit your budget better, but, be sure you know what capabilities you're trading off.

Make sure you incorporate current med-forms on the boys and adults on the trip. We usually ask parents if it's ok to keep the most recent summer camp med form on file when we take trips, and nobody has ever said no. I'm sure there's some HIPA law we're getting in trouble with...

Also, it's great that you have a doctor with you on your trips sometimes, but, spending money on at least WFA level training for your other adult leaders is always worthwhile. In addition to whatever money you spend on the FAK, make sure you budget annual restock money for the FAK, it's not a one shot purchase, and recertification training money for CPR and whatever level of First Aid training the leaders need to hold.