Dyce51, it sounds like the people/places you talked to about using their kitchen need a little more information. I doubt they even asked their insurer, but even if they did, you would carry your own insurance and add the business as an "Additional Insured" to the policy. It's a common practice. Happens all the time in the construction industry, and i've seen on some festival applications that they require the vendor to add the festival as an "additional insured".
And the FDA does not need to inspect the place. You would most likely be licensed under your city, county or state health authority, depending on where the kitchen is located and what your state regs are for wholesaling-vs-retailing. The FDA has nothing to do with small processors like us until you get really big. In fact, you don't need a nutrition panel on your label (as long as you're not claiming things like Low Calorie, Fat Free, Low Sodium etc) as long as you're making/selling less than 20,000 units a year.
Where I live, if I'm only wholesaling, I get licensed through Washington State Department of Agriculture. If I'm only retailing, I get licensed through the county health district, the same people who inspect restaurants and stores. If I both retail and wholesale, I have to get licensed from BOTH agencies. (what a crock!) That's pretty much why I only wholesale, cuz the county ticked me off and I refuse to buy another license. It wouldn't be so bad if all the counties required county health permits, but that's not the case. Every other county in the state accepts the state license to allow retailing, only our county requires a county permit to retail.
Anyway, good luck with your search. You'll find something that will work.
edit- look at places like grange hall, fraternal lodges (Eagles, elks, masonic, etc) maybe churches or private schools???
SL