Can you remove the redirect from transactional replication from an HADR publisher?
Apparently I was overthinking the problem. Running the following:

Exec dbo.sp_redirect_publisher @original_publisher   =  'SQLServer03',
                               @publisher_db         =  'LMDB25',
                               @redirected_publisher =  NULL;
Go


where @original_publisher is the currently and original primary node,
and @publisher_db is the database name.

To replace the redirects so that replication will work normally again simply re-run the command changing the NULL parameter value to the server node name of the previously offline server node:

Exec dbo.sp_redirect_publisher @original_publisher   =  'SQLServer03',
                               @publisher_db         =  'LMDB25',
                               @redirected_publisher =  'SQLServer04';
Go