If you find a way to clean a chain on the bike without getting messy, let me know.
Back in the old days with master link clips, one could just remove the chain and soak it in a pan of kerosene and scrub it clean, dry it and lube it, then reinstall.
(Actually before O-ring chains, you would heat the chain in a pan of kerosene to have it penetrate into the rollers)
But not with riveted master links these days.
You might be able to find some matting to throw down under the swingarm with some absorbent towels to prevent staining the floor, but you may just have to wait until spring with the weather warms up and roll it outside.