Hi. Welcome to Just Answer.
At minimum you will need the car battery. This should power the pump for about 15 minuets if that.
The inverter is designed to run off of a system that is connected to a generator/alternator. The average inverter will drain a car battery in about 15 minuets running a small appliance from the inverter. If you hook up a microwave to it you will be lucky to warm a cup of coffee.
The pump size is adiquate for what you want to do. At 22.5 seconds it will put out 2-1/16 gallons. The size of the apperature that it is pumping it through will be a factor though. At 25 - 30 feet output at that flow rate a 3/8" nozzle would be pushing it. A 1/4" nozzle would restrict the pump (causing more strain on the inverter/battery) and you will not get the GPM mentioned above.
Best suggestion is to rent a small generator from your local rental yard to power the pump.
The inverter and just a battery will not cut it. Figure a 20 minuet parade march you would be out of battery at the half way mark.
With a small generator you will probably run out of water first.
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Steve, I am most intrigued with your plans. Please let me know how it turns out.