I suspect there's a sizing issue here because I'd be surprised if a 3000W inverter could start a 3000W motor.
If you run the inverter at its nameplate capacity, your immediate 12 VDC current draw will be 300 amps or more. Aside from inverter losses, you will have considerable voltage drop in whatever wires you are using to connect to the batteries, and indeed, through the batteries themselves. The inverter will compensate for voltage sag by pulling more current.
So that means you'll need 300AH per hour. 5 AH per minute.
Invert as soon as possible ... For the big loads
You'll want the inverter as close to the batteries as safety permits, with battery-battery and battery-inverter interconnects of heavy 2/0 or even 4/0 wire. I've seen a solar project fail because they used (seemingly generous) #10 wire to carry 20A@12V eighty feet.
The goal is to cover distance at the highest voltage possible, because raising voltage both lowers current and lowers the importance of voltage drop. The benefit is (voltage/voltage) squared - e.g (120/12)^2 is 100x better to transmit as 120V than 12V, and (240/12)^2 400x better for 240V.
With 120V you must also watch voltage drop, but it becomes a factor over much longer distances in the 100-200' range (and don't forget the distance down the well, unless the motor is at the top; most are at the bottom).
Use a voltage drop calculator with your distance, voltage and continuous current draw (not startup surge). Override the absurd 3% and allow 40% or so, merely to see how bad it will be at the minimum allowed wire size. Anything more than about 6-8% needs remediation.
- You can recompute with a 6% value and see what it says for wire size.
- If it's recommending hundreds of dollars of wire, recompute for 240V. If that number is much more favorable, then it's time to go 240V on the well pump (but do not increase horsepower!)
You don't need a 240V inverter. You can make 240V (at half the current) with 120V and a simple transformer. The 120V side of the transformer looks like any other 120V load. A motor of same horsepower will have the same current draw (from the 120V side). Transformers have a trivial power drop of 1-3%. They handle startup surge without worry (unlike inverters) so size for the motor's continuous draw.
Deep cycling is a problem, though
As discussed, you'll need 300A+ from a 12V battery. That's 5AH per minute or 300AH per hour.
Beware lead-acid, as they take damage from discharging, so their practical, everyday capacity is nowhere near their nameplate capacity. Seriously. If you draw only 20-30% of a lead-acid's capacity and then recharge it, the battery will work for years. If you start dipping it to 50%, you shorten its life to a few hundred such dips, and if you discharge 80% of its nameplate capacity, you'll greatly shorten its life to dozens of such cycles.
So a 300AH lead-acid battery would very quickly fail. Best practice is a 4x oversize, so now you are talking about a 1200AH battery.
Lithium batteries do much better, and nickel batteries are almost impervious. Apple and Tesla get phenomenal performance out of lithium batteries (this 7yo iPad is run down daily and still has nearly full range) by including 20-30% more battery and not letting you use it. Once Tesla pushed a software patch to hurricane customers that let them use the full range of their battery. This gave them 100+ more miles!
Make hay while the sun shines
If you have a location at a higher altitude than your house, you can put a storage tank there. Water in the storage tank pressurizes your system and delivers water to the house. The pump tops up this tank when the sun is shining and the panel is at peak delivery.
Faucet pressure is based on altitude difference, 1 PSI per 2 feet of altitude.
So for instance if the solar panels are producing 2000W of power, your 3000W load is only drawing 1000W from the battery, which has a big impact on sizing.
If the solar array is 3000W then the battery sizing issue pretty much goes away (least as far as the pump is concerned), you're only borrowing from the battery for startup surge.
As a bonus to this, you can spin down the inverter when you're not using the pump, so you don't have the parasitic "standby losses" of the inverter, which really beat you up because they're 24x7.
Also, this lets you do irrigation at sensible times rather than at the height of the day when the sun is shining. The tank size is the limit; essentially you're storing "pumping energy" hydraulically rather than electrically. Tanks are cheaper than batteries.