Is it possible to have both of them?
Yes. They can be circumvented.
However
i.e. version 1.0 in production mode and version 1.1 in beta mode?
It prioritizes the larger version code. So if your version code is in versions 1.0 and 1.1 in beta, then what you see in the Play Store will be in beta. Regardless of whether it is Open Beta Testing or Community . Didn't try Closed Beta
Or should I support another beta application (which is not convenient as I need to change the package names)
In my case, I already have the application in Production, until I decided to add products to the application, so I downloaded the version of the application in beta. It turns out that he has the same package name as in the Play Store. What I did was deactivate the product first and disable the beta. When the application is ready, upload it to Production and activate the product.
As an additional thought on versionCode let me share this answer
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