Each 600W lamp draws 5 Amps @ 120V, so that's 10A right there. Add in another 10A for peripherals, and you're up to 20A. Will you be running air conditioning? Dehumidifier? Those things add to that amperage total.
Let's assume you're not running AC or dehumidifier, and will only be running the bare minimum, so we're back to 20A.
To calculate the generator output, we take 20A and multiply it by voltage: 120V x 20A = 2400W.
You will need at minimum a generator that has a continuous run wattage of 2400W. I would at least double that in case you do need to expand later.
Now, have you considered the noise and fuel cost impacts? What happens when the generator needs service? What happens when it breaks down?
I have a 10kW generator that I use for my grow operation (as well as the majority of my house) when power is out for long periods. I also have a decent sized solar array for when power is out for shorter periods, but I wouldn't run a grow solely on a generator.