Well, Carson was an Ulster unionist who opposed Irish home rule, agitated for a government in the North if home rule was granted, and set up the armed paramilitary Ulster Volunteer Force.

The phrase 'No surrender' goes back to the seige of (london)derry where the Protestant loyalist city refused to surrender to the Catholic Republicans (The seige was lifted when William of Orange despatched ships to break the blockade, hence the advent of Orangemen and 'Orangeism')

Hence the rat would be something or someone with Irish nationalist sympathies or tendencies.

EDIT - so it makes sense