Why do well pumps need 240V when the wattage is not that high?
Need is not the right word here.  Preferred is probably a better choice.  240V is preferred because submersible pumps are often well away from the house and deep down in the hole and you must run a wire there capable of supplying its power needs without too much voltage drop in the line.
While you could do this with a large AWG wire at 120V, you can use a smaller and less expensive wire at 240V.  So this is commonly done.
I've not shopped for one but I suspect you may be able to locate a 120V submersible pump if you needed one.  By the way, keep in mind that a 500W pump may need a 1500W inverter to start it up.  Check the specs for your specific pump.
Apart from that, why not get a 240V inverter for your battery backup instead of a 120V one?