Passports for children of EU citizens living in NI

[threedee]

Hello.
Not even sure where to begin.
Me and my wife are EU citizens but our kids were born in Northern Ireland and now at a point where we need to figure out passports for kids for travel. Decided to attempt Irish passports as our own country doesn't allow for dual citizenship and my sister is UK citizen so, if successful, we will keep EU connection across border after brexit. The whole thing is long overdue as well.

Some facts:
Me and my wife are residents in NI since 22 July 2004, married in NI 2007. We both hold Lithuanian passports. There are breaks in employment due to 2007/8 economic crisis.
Kids born on 09-06-2007 and 14-03-2010 and have no passports.

We are especially worried about our first born as it appears if she is born too early by a couple of months.

We've already sent in the forms with earliest proofs of our employment (P60 2004-2005).
We dont have the proof of our EXACT arrival to NI as there were issues (on employers side). I do have however obtained a signed letter on a company letterhead of my first employer as proof of my employment there (went there to get it several months ago just for this whole passport thing).

What are my options ? What else can i do to ensure passports for my kids ?