Likewise, nobody will be all that surprised, or disappointed, if Binface never takes his seat. It's much more of a protest vote than voting Sinn Féin in Northern Ireland, and voters will have achieved their aim of Farage not getting in.
Actually, I'd take issue with describing Sinn Féin as a protest vote at all. They've historically been the only choice that even claim to represent constituents in many areas. And they do seem to do much of the work of an MP (writing letters on behalf of constituents, lobbying government agencies...) they just don't vote or debate.
Agree that SF is not a protest vote but they do vote and debate. Since their first major electoral victory over 100 years ago they have been clear that they represent their constituents to the best of their abilities (I won't opine on the quality of representation in this forum). That has always involved a cabinet, votes, debates, and eventually a bicameral legislature.
The key aspect is that they consider the English government to be a foreign government and so they avoid involving it in the work that they do in Ireland for their Irish constituency. Statements about the illegitimacy of their government historically come from conservative English sources. But the fact is those SF debates in their "protest government" formed the foundation of the modern Irish state. They are a protest vote in the same way that the US Constitutional Convention was.
By the same token, they consider legislating on affairs that pertain to the English, Welsh, and Scots to be none of their business. To take up seats in a foreign parliament would be to meddle in the affairs of a foreign, sovereign nation. And that would be hypocrisy!
The thing is, an SF MP can't vote or debate because they, by policy, refuse to attend. You're correct that the party has representatives in the other bodies which it does recognise but those aren't the same people.
So for example Órfhlaith Begley was elected MP for West Tyrone. Her voters will have known she's not going to Westminster, and she didn't - but she's not in some parallel institution instead, AFAIK there isn't one. Nicola Brogan represents West Tyrone as an MLA in Stormont, because Sinn Féin does recognise the Assembly and you can't say well but there's a body in Dublin. Dublin doesn't control West Tyrone so what would she even do there?
You're absolutely right that they aren't in a parallel institution. As I understand it, the SF perspective is that there is no legitimate government representing the people of West Tyrone (other than Stormont, which is complicated). I think it sounds like we're saying the same thing but there's a subtle interpretation difference. When Idi Amin declared himself to be the King of Scotland the Scots didn't send representatives to Uganda to form a government. The SF position since their Árd Fheis before the 1919 elect has been to treat Westminster like Idi Amin. To simply engage in a conversation about Idi Amin's claim to the throne of Scotland would be to give him too much credibility.
So the core issue, from the Republican perspective, is that the people of West Tyrone are denied representation at the national level in Ireland by a foreign government. Once that representation is achieved the SF representatives will participate in it. (again, not trying to address the merits; just clarify the logic)
But Órfhlaith Begley does go to Westminster, and she has an access badge for the Houses of Parliament, and she flies over there and works in her office in the parliament buildings and answers her @parliament.co.uk emails and asks her staffers (paid for by the parliament) to respond to letters on paper headed Órfhlaith Begley MP. All the work of an MP except the most performative part.
She just doesn't go into the room which is called the House of Commons and try to speak or vote there, because the armed guards at the door won't let in anyone who doesn't swear allegiance to the King. If swearing allegiance to the King was a requirement to use the email system, then she wouldn't do that either.
Fascinating, I didn't know that they actually physically travel to Westminster. That does seem like quite a trip given they could work from home. I'm also not sure how effective you can be in this way, but of course it's really up to her constituents not me whether they feel adequately represented.
I think it's parliament.uk rather than parliament.co.uk by the way.