2nd button start, hmm. Based on that, the lowest (chinmost) of the helper buttons would be ideally placed for a C-nat pull. The highest (kneemost) of the helper buttons would be ideally placed for a high G# push.
What you would do with all the other options would be a bit of a puzzle. G# on the lowest button would be a bit out of the way, but usable I suppose. (And still lot less out of the way than accidentals on many D/G systems!)
You could pair a pull F-nat or F# nicely with the push G# on the highest button. What you would do with the middle button I don't know. Maybe E/D as handy reversals.
Or go with my original suggestion and put up with the fact that the notes on the lowest two buttons at least would not be ideally positioned.
Maybe someone else has some ideas!
BTW for Irish music, the most useful single note is the C-nat. Then it'd be a toss-up between F-nat and G#, depending on whether you would prefer to be able to play tunes in Dm/Gmix or A major respectively. Playing in A major would definitely be the easier choice, trust me