A flat rate fits for a fixed price, (going for a flat rate "of price" implied). Ex. Every letters are sent at flat postal rates. (that's untrue, hey!)
A flat fee fits for a fixed rate of fees. Ex. Flat landing fees for light aircrafts.
Let's suppose fee rates are at stake, for sure you might say either way: flat fee or flat rate. But rates are not always rates of fees. Rates are sometimes rates of something other than fees, for instance prices or assessments. So I guess it's a matter of context. Just my own opinion. It should be checked further.
This subject of money, fees and rates is always very complicated! Thanks for your confusing explications, you did your best for it.