Irish protestants are right-wing.
not socially
The Irish, in general, were and are more right-wing and conservative than most Europeans. Irish Protestants vs Irish Catholic is complex because as you say, in some ways the Irish Catholics are more typically conservative but Irish Protestants are more pro-power/security/establishment in ways that the Irish Catholics are not. It depends what aspect of right-wing we are talking about.
In terms of LGBTQ, they were both very conservative to begin with but the Protestants have been more open to it within Irish culture. In terms of substance abuse, the Catholics are more lenient on alcohol abuse, even celebrating it as part of their culture while the protestants can be but are usually a bit more conservative about it all. The ones that party the hardest and have the typical 'Irish furious drunk' types at their parties are typically the Irish Catholics and that's not socially conservative. Irish Catholics are also known (this is just word of mouth but the birth rates do suggest it) to be more promiscuous than the Irish Protestants, this is closely linked to their alcoholic partying lifestyle.