Is it OK to have a fire pump main service disconnect then feeder to the fire pump controller which has it's own disconnect?
Too many disconnects.
In this case the fire pump is far from the electric service (>350'). The fire pump service entrance could be run all the way to the fire pump room. However the standby generator is located at the electric service so it makes much more sense to put the main disconnect and transfer switch near the electric service (and generator) and run a feeder to the fire pump controller.

It will be best to check with the design professional for the project, and the local AHJs, both Electrical and Fire

Read thru 695.4 (B) (1). IMHO, there are too many variables to say yes or no to your situation.

I see nothing in the NEC that would prevent you from having two disconnects, in series, for a fire pump installation.
I have designed, and had installed, new disconnects with short circuit protection upstream from existing fire pump services. They have not been questioned by the AHJ.