Hi Too_Sleepy,
Unfortunately I think you will find it difficult to get a clear cut answer here to your query. Your best bet is to contact both authorities with your proposed training plans and make sure you have any agreement to accept them, if forthcoming, in writing.
There has been a good understanding between the IAA and CAA in the past but I believe things have tightened up with the IAA in more recent times with the availability of commercial training available in Ireland. Sorry I dont have any first hand experience myself.
Regards
Willby