So that doesn't exempt this from the laws of physics.
Using fuses as the running OL protection is tricky to pull off. 125% is the max you can use given the SF, but if the fuses are not time delay, the starting current may blow them. Even if they are time delay, it's often a craps shoot because you don't know the total starting current / torque/speed curve of the motor as applied to this pump used in this way. For example in your description, I've seen people not understand that if the output of the pump is just connected to a fire hose or huge pipe with no head pressure or flow restriction, the pump will go "open channel flow" and can easily overload. An OL relay may give you enough of a delayed reaction to allow the pipe or hose to fill, but that's really hard to pull off with fuses. They are not really made to be used this way.
A better plan, even if they don't want a contactor and controls, would be to repurpose the fuses and disconnect switch somewhere else and use an enclosed Manual Motor Starter, since you already have the Short Circuit protection with the up stream breaker.
But before spending any more $$ on hardware, take current measurements.