Can I use regular gas (stove) cookware (pots and pans) in a fireplace?

It's not impossible, but you should choose your pans with care. That 550°C figure is for the outside of a wood-burning stove, and would be faintly glowing red in a dark room. In practice the stove has to be burning pretty fiercely to get that hot. You could put pretty much any cooking pot on top and you'd be fine.

But you say "in a fireplace", not "on a wood-burner" so it will get a lot hotter than 550°C. It's done when camping, but it's also how many people cook all the time - the 3-stone fire (google images) is very common, though not very efficient. You usually need a sturdy pot support as resting it on burning wood isn't good for the pan or the food, especially when things move, hence the three stones. Cooking in the embers is possible, but that's normally done by letting them die down, and partially burying the pan for slow cooking.

Cast iron pans have been used for cooking since before gas stoves were invented, so they'd be fine. My cast iron is enamelled and I wouldn't because I want to keep it looking nice. Also the hottest burning bits of the actual fire are potentially a bit too hot for the enamel.

A pan full of hot water or milk is limited in how hot it can get, which will protect many pans. That means that thin aluminium camping pans can be used right down in the fire, but not if they boil dry. They don't last for ever if you make a habit of it, but they do work.

Given what I have available, I'd use a stainless steel pan with a stainless steel handle and lid. Obviously the handle has a high chance of getting hot enough to burn you.