First, these are copper-PLATED steel jackets. There's a huge difference in both the process and appearance of copper-WASHED bullets and copper-PLATED bullets.
Second, the rifling of the barrel will dig much deeper than the couple of micron thickness of the copper plating. This means that the sharp corners of the rifling will be engaging steel instead of copper. It should make sense that mild steel is still much harder and tougher than copper, and will put more wear on the rifling of the barrel. You aren't going to wear out your barrel in 10 rounds or anything, but there is likely to be some increase in wear.
Now, since you have a stainless barrel, I'm guessing that this is a match-quality barrel (if it isn't, then IMO you would have been better off with a chrome-lined barrel). If that's the case, then we have a couple of problems.
A barrel only has a limited lifespan where it will retain its highest level of accuracy. This will usually be somewhere in the neighborhood of 3-8,000 rounds, depending on a number of factors including how fast you shoot. Beyond this number of rounds, groups will start to open up, and the only way to get them back down is to replace the barrel.
1. Why would you shoot steel-jacketed ammo in this barrrel, knowing that you're accelerating wear on the barrel, and
2. Why are you not shooting MATCH ammo out of your MATCH rifle? That's like putting 87 octane in a race car and wondering why the engine isn't running right and you have much less horsepower.
A racecar, with its race-tuned engine, open exhaust, soft, sticky tires, and high-octane fuel works together as a SYSTEM. So does your rifle, mags, and ammo. Don't be surprised if you get poor results when you change the component parts of a system...