Is it possible to include a second app icon to be used on macOS versions prior to 11.0 Big Sur?

No. You can change your DockTile dynamically, while the app is running. So you could ship a regular icon, but show a rectangle version while running on Big Sur.

I noticed Big Sur default apps have the old icon (Catalina) in the resources. This icon is present both in the icns form and in the Assets.car. Also in this Assets.car file is present the new and rounded icon, that shows in Big Sur. If you copy/paste one of those apps to Catalina you'll see the old icon though.

So it might be possible and makes sense they give us this possibility.

Just add AppIcon-1016.icns file in the app bundle.

See Steve Troughton-Smith's discovery here: twitter.com/stroughtonsmith/status/1278364051572232192 (can't make it a link: "This URL can't be included in your post"…).

@vslavik: While this works, wouldn't it mean that the Mac App Store still shows the old icon, though?

This solution ceased to work with the latest beta. Does anyone know an alternative? I really don't want to display my Big Sur app icons in previous macOS versions. Changing the DockTile is suboptimal as it only works after the app has launched, of course.

I'm also wanting to do this and the previous method seems to have been removed.
I have filed a Feedback suggestion report FB8768688 and suggest everyone else does too.