Before you decide on anything, try adding some weight to your K-80. When I shoot my tubed guns I use a Kohler barrel weight to compensate for the tube weight when they are removed. I find the additional weight beneficial as it smooths my swing and reduced kick, but I'm not petite. You may, or may not like the heavier gun. You won't know until you give it a try.
There is a benefit with a carrier barrel that has nothing to do with actual shooting: You don't have to do a barrel cleaning between events. This is a real factor when it comes to competition shoot-offs. If you have to shoot a small gauge, immediately after the 12-ga, you'll be in a rush to get your tubes in. It's not a problem with a carrier barrel set-up.
Some shooters find they actually get better scores with their 20-ga tubes than they do with the 12-ga. They seldom shoot the gun without tubes. Other shooters prefer a soft kicking auto in 12-ga events.
If you're like many others, you may find yourself practicing mostly with the smaller gauges as they help fine tune your skills. And there's less recoil, also .410 reloads eat up shot a lot slower than 12s. There is one disadvantage to shooting the small guns exclusively in practice, when you go back to the 12-ga, the added recoil can be a surprise at first.
I wouldn't consider a stand alone .410 because they are just to whippy. They are quick to get on target, but as quick to loose it. There is a myth among new shooters that you need a light gun to quickly get on quick target. With experience you'll find the target aren't really that fast and a smooth swing (from a heavier gun) is more important.