I’m from CA too. There no reason why you theoretically can’t shoot both. It won’t do any harm, but you need to know the differences and treat them accordingly. With a thorough cleaning, you can change up between lead or any of the monos without any issues.
From an accuracy standpoint it is highly unlikely any different bullet will shoot the same, especially copper vs lead. I’ve seen 100 yard differences between the two that are an inch or so, and I’ve seen them 10”. So it’s pretty hard to sight in to hunt with copper and then practice with lead. Best to practice with what the gun is sighted in for.
Additionally, since you posted this in the reloading forum, I assume you handload. Pay VERY close attention to load manuals as work up slowly and carefully. NEVER substitute a copper mono bullet for a recipe that uses a lead bullet. Copper will show pressure much sooner in most cases, assuming same bullet weight. Another way to look at it is copper acts like a much heavier lead bullet. You can’t just swap them into recipes interchangeably.
As for Barnes v GMX v Etip. You just won’t ever know until you try them. I have guns that love a TTSX, yet hate a GMX, and vice versa. There’s no substitution for trying them out on your own. On game, I find them equally effective. No preference and I’ve shot them all, a lot. Put any one of em in the right place and they all work fine. Go with whatever is accurate.