As for "fancy" copper or other bullets, for most anything a .44 carbine is good for, any good quality for caliber weight jacketed or cast is all you need.
Unless you live in / wish to hunt in one of the states that has (or will) outlaw lead for hunting.
Being aware of the hazards of lead is a good thing. Our current "lead-phobia" is not.
Don't know if they still do, or not (not being likely) but regarding hazardous materials, they used to teach us a quote, from "the Renaissance alchemist Paracelsus" about how everything was poison, what varied was the dosage.
Lack of understanding that, being applied in the real, not the theoretical, world is the root cause of many of our problems today.
If some is bad, then less must be good, and none is the best. And its opposite, if some is good, then more must be better are both being applied to about everything in our lives, and neither is entirely accurate.
Nor intelligent, when applied with a "zero tolerance" approach.
DOSAGE (exposure) is the key, and the single most important factor in determining harm or benefit.
EVERYTHING in our world is "poison" if you get too much, and a great many things are harmful to us ("poison") if you don't get enough. People who don't think this through often try to rebut the idea using something common in our lives that isn't (normally) "poison" as their example.
Many choose water, and say "water isn't poison!" To which I reply, "well what are your odds if I drop you in the middle of the North Atlantic (in April)??
DOSE (Amount makes all the difference) AND the harmful amount is VERY specific to the material. A few milligrams of lead in one chemical form can cause illness, injury, death (lead poisoning)
Many, Many, Many times that amount in a different chemical form doesn't. I've known people who lived half a century + with a lead bullet in their body, very close to vital organs. After they survived he "injection" (getting shot) the lead bullet did no further harm.
Back to bullets for a .44 rifle, the "right" one is one that fits your rifle bore and gives good accuracy, and isn't too long or too short to fit and feed through the magazine and action.
Personally I stay away from the really heavy slugs, 240-250gr had always done everything I asked, and expected them to. for me, 300gr and + is just an unneeded complication, assuming I can get them to work at all in the first place.