You might want to contact one of liability insurances that provide coverage here in the US and see what other options/suggestions they have. I would think that since you are providing classes/services within that base (I’m assuming this is the case even though you’ve mentioned NATO base), same laws should apply as if you were working on US soil. If not, you might want to check in with the company that has contracted you to see if they could offer some type of coverage for you through them. Have you asked other instructors there where do they get their coverage from (if they have one)? Government contractors usually are covered by the agency which they have been contracted with for any type of operations/services they provide no matter the location.
I know there are fitness professionals in this forum who are in the same situation as you are and it would be interesting to see what they have to suggest.