The British bailed out the Irish economy, so why do Republicans think that cuts are the way forwards?

There's a reason countries don't want to get bailed out, though. The lenders usually impose those kinds of budget cuts anyway because they understandably don't want to see their money disappear. Having your domestic tax/spending policy dictated by a foreign entity is usually not an endearing plan for democracies. Look at IMF policies towards countries like Indonesia in the 80's, or EU demands on Greece last year.