I have battery back up that powers my well pump. But, I also have solar panels so this is part of that whole system. You can get the inverter and batteries now and add solar panels in a couple years. So then your well pump back up is just stage one of a larger system.
First, I changed well pumps when I went solar as my original well pump was 240V, so my well pump is 120V and either 1/2 or 3/4 hp, I forget. I have a top of the line inverter/charger controller like this
https://realgoods.com/pre-wired-schneider-conext-sw4048-off-grid-inverter-system-w-midnite-classic-150-controller and while that may look expensive it is the most expensive part of a whole house system and has lasted 20 years so far, this brand, more or less, this brand is who bought out the brand I have. An inverter like this is all ready to both be grid tie and to wire to back up batteries, all that is built in, no other components to buy, just wire batteries and panels. Everything that powers my house can be bought for $10k-ish, total ( I only have 2. 5kW of panels, it is not true that a house needs alot of power).
You can absolutely buy a less expensive inverter to power your well, there are others that can handle a short start up surge.
you can call the technicians at real goods and they can advise you, I just think you should think ahead as to what you want over all, just well pump back up or solar in the future ?