Two big problems here: government won’t ask for correspondence and government won’t answer questions about appointment because of the separation of powers. Both these stances are misguided. Appointment and removal are the only checks on the judiciary... https://www.irishtimes.com/news/politics/government-will-not-seek-rest-of-correspondence-between-woulfe-and-clarke-1.4410028?mode=amp
Removal is an expressly legislative function and is this correspondence is relevant to it of course the legislature can ask for it. Compelling it, particularly if related to the administration of justice, might raise issues, but the idea that it can’t be requested seems wrong
Appointment of judges is a government function. The government is accountable to the legislature for the performance of its functions. Of course it can and should answer questions about it. Neither of these things would undermine the separation of powers or judicial independence.
Constitutional checks and balances like removal and government accountability for appoints are part of the separation of powers, not threats to it. Respect for judicial independence is great, but it can’t cancel out other crucial aspects of our constitutional order.