Jumping-in with my 2-cents...
I agree that power is the big issue. If this "show" is in a theater, the power will be available, and there's probaly a "lighting guy" that can help.
Otherwise, it's a "call an electrician" to run temporary power-lines.
Or, "plan-B". Maybe ~1000W total. My dance-lighting system uses eight 100W colored floods (plus some lower-powered effects and the sound system.) Since it's typically all on one circuit, that's about as high as I want to go.
Or, a xenon strobe (or multiple synchronized strobes) would require less power. The strobe "charges" for a short period of time and then flashes for an even-shorter period of time. So, the instantanious power into the flash-tube is higher than the maximum power drawn from "the wall".
Sure it's hard on the lamps, but I'm not so concerned with you "wearing them out". If you go to a concert, the lights are constantly switching on & off. My 100W lamps are pretty active too. I don't use my set-upvery often, but I've had the set-up for several years and I've never lost a lamp! With more than one lamp you've got redundancy, but it wouldn't hurt to have a spare bulb on-hand.
I also like the solid-state relay. It's simple, and those things are very rugged & reliable as long as you don't overload them.
And if you go with 4000W, I'd use a separate relay for each lamp. It will make things easier for you and the electrician.
There are a couple of issues with mechanical relays - A high-power relay is generally going to need more power to the coil than the Arduino can supply. That means a 2-stage set-up with a MOSFET & power supply driving the relay.
Secondly, mechanical relays are slow. If you are thinking about using a mechanical relay you can check the specs, but I'd guess that a high-power relay can switch about 5 times per second. It would probably "work", but you might have to compensate for the time-lag. I'd "feel better" if the relay can switch at 10 times the necessary speed (10 times in 1/2 second).
On the other hand, a solid-state relay can switch in a few milliseconds. Then, they will generally stay-on 'till the AC power goes through a zero-crossing. So at 50 Hz, you can't turn on & off faster than 100 times per second. (Two zero-crossings per cycle... Still much faster than an incandescent filament can react.)