But maybe there is an elegant way to "import" a cluster?
It should possible to accomplish your goal without performing "initdb" - though you may find it easier to forgo the Ubuntu utilities and just write the necessary pg_ctl commands yourself. You indeed need to use either "$PGDATA" or explicit command line parameters and just point directly at the cluster data directory that you mounted - with pg_ctl, not initdb.
Whether that is easier than a simple pg_dump/pg_restore "import" it up for you to decide.
You might be able to rename things to match the expected structure for the Ubuntu cluster commands.