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.