Backup Oracle Data Guard with TDPO
When using Oracle Data Guard with TDPO, what is the best strategy to
  ensure backups continue after failover/switchover?


Use RMAN catalog, because without a catalog, backups taken on one site do not appear in the controlfile of the database on the other site. You need a centralized catalog for backups in a Data Guard configuration.

Use role-based services, and TNS entries with connect-time failover, using these services. Use these TNS entries for logging in to the database when performing backups. This way you always log in to the primary/standby database, and you do not need to check for the role of the database.


  TSM/TDPO uses a feature called datamove which makes sure that backups
  from primary and standby are moved to one TSM node. (TSM node A
  (primary) and B (standby) move there Backups to C. Restore will use
  C).


The database/RMAN layer has nothing to do with. Just configure TSM and TDPO properly, and this will be transparent for the database, always using the same tdpo.opt file.


  The RMAN scripts called by TDPO need a logic to prevent backups are
  taken from both (primary and standby). This could be done by querying
  the role of the database.


No, see above, role-based services, connect-time failover TNS entries.