That's hardly a PostgreSQL issue. If your container tech does not allow installing both old and new version of the binaries, it's a silly container tech.
Why not? Having the whole application in one Kubernetes setup, with all databases and other services is very convenient. Are you talking about performance overhead, or are there other drawbacks to this approach?
The main problem is "persistent storage". In k8s, a PVC/PV is probably close to the right answer, but in general that gives you a file-system view of the storage, not a block-device view, which for some databases might matter).
Also, in general, that persistent storage is (generally) not locally attached, which means that (some) file system interactions may have unexpected semantics. Now, with a DB, you're probably only going to have the PV attached on a single worker node, so most of that will actually just work.