Quantity

600W HPS is fine for a 4x4 but as you said heat could be a issue with HPS especially in summer, i you can afford some LED you should look into this:

4X4 - Spider Farmer Official (spider-farmer.com)

this are the model from spiderfarmer that would do a great job with a 4x4 but there some equivalent with some other brand, like viparspectra, phlizon, mars hydro etc.

I am with Vince on this even if you get the hps for free, led will serve  you better in the long run.

I use viparspectra lights and one ac infinity light. They are all very reasonably priced and the heat from them is easily removed from the grow area. If you do use hps, I prefer the ones that have the fan air flow through, like tube lights. And a place to vent outside the room

If you shop around you can find  led's used. The local Craigs list here has 4 HLG Diablo leds for 200 a piece and that listing has been up for a while, so he would I suspect, take less to get rid of them. You get the idea