And this guy still wants it to have one-minute sustained output. He hasn't provided anything that I consider a compelling reason for this.

Lasers with a power this high only come pulsed as of now. Instead of continually outputting it "collects" and gives the radiation in short pulses (up to femtoseconds - 10^-15s), which, if you calculate the output (Energy over time), gives you the high numbers.

Romulans