The correct answer will depend on what what you mean by a read-only connection.

Setting the ReadOnly DBI attribute with a DBD::mysql database handle doesn't do anything useful; it's just advisory as explained in the section you linked.

If you want to check if the database is read-only (as is common on a slave), the user only has select priviledges or if the transaction is read-only then you will need to write the appropriate queries to figure those out.