E26,
With the type of controller you have, you will damage your inverter and pump if you don't have a battery in your system. If you are going to run daylight hours only, almost any battery will work. It is there to stabilize the voltage. Without it, the voltage will spike and sag with the solar output and inverter load. If you are going to run at night, I would go with the golf cart style 6 volt batteries as needed for the load you anticipate (two batteries minimum obviously).
Let us know how it goes after Friday's upgrades!