I'm going to initially test the super capacitor bank by itself through a 24V split phase LF inverter. One thing I've read is to be careful the capacitor bank is the same voltage as the batteries. You could damage a battery otherwise.
You need to make sure the bank is the same (or very close to) voltage when you connect it, if it isn't, it will VERY RAPIDLY try to make itself the same voltage.
The actual capacitor bank can support as high of voltage as you want though, say that 48v capacitor module in the link I posted, it just won't be using any of the upper limit of voltage if connected to a 12v or 24v system and will probably be wasting capacitor capacity (say it really fast over and over). Once you install it inline with the rest of your system it doesn't have the ability to get voltage higher than what flows through it.
Of course precharge capacitors through resistor before paralleling with battery.
Total energy in capacitor is 1/2 C V^2, but the energy available to inverter is difference between that energy when V = initial battery voltage minus that energy when V = low voltage disconnect of inverter.
Capacitor has a series impedance (resistance), and battery does too. That determines current sharing when inverter draws voltage lower, and voltage vs. current curves determine how much current is available.
Lithium battery can probably supply all the current inverter needs to start any motor. BMS maybe not. A system that bypassed BMS FETs and had BMS wired to remote on/off of inverter, if it provides suitable delay, could work.
What is surge of pump? If 40A 240V, about 10kW, from 24V battery that's about 400A. A 2C surge for one second from 200Ah battery? That would seem something it could handle. Maybe just get a BMS with specs to cover that.
I've watched some super capacitor tests and they often don't actually have the nominal capacitance. People with huge battery banks don 't need them, but if I can get an extra 1.3A @ 230V for 10 seconds, I'll take it. The cost was about $120.
An extra 250W won't do anything for you.
But if delivered in 0.5 seconds, an extra 5kW is probably what you need.
The nominal value is 500F for each capacitor, but people have tested these as being lower, about 350F. So 12 in series is only about 30F. I am using 30V as the level of charge (2.5V ea)
30F, drained from 28V to 24V would deliver 3120 Joules or Watt-seconds, according to my math.
So 6kW for 1/2 second. That could be useful.
But look up resistance, and if possible impedance at 60 Hz (ripple current drawn from battery.)
Determine how much current will be drawn from battery vs. capacitor.