Roundhead? I think you might have the wrong historical pejorative, there.
Roundheads were Parliamentarians, supporters of Cromwell, Republican and “Protestant,” being mostly Puritan, Presbyterian, Baptist, etc. although some were CoE and found being so described offensive. Some still do.
Cavaliers were Royalists and almost entirely CoE. Then there were the Catholics, allied with Royalists of course, since the whole kerfluffle initially centered upon the Catholicising tendencies, real or imagined, of King Charles I.
So, the historical pejorative you’re searching for would be “papist.”