Unfortunately, they (the IAA) are right. It may be a JAA approved PPL school that you are doing your training with in the US but it is JAA approved by the UK CAA. You are therefore deemed to be completing your training under their supervision. You'll still be issued a JAA PPL which you can use to fly in any JAA member state or on any JAA member state registered aircraft, including Ireland.
I know, it's a pain in the tits! When I did my PPL, I had about 20 hours logged here and had completed all my ground exams when I started getting seriously frustrated with the weather and amount of cash that I was spending. I looked at the option of going Stateside to complete my training and sit my skills test. However, as the IAA don't approve any flight schools in America, I would have had to switch over to the CAA and re-sit all my ground exams. In the end I decided it was more hassle than it was worth and completed my PPL here.
If you've already completed most of your training in the States and you need to resit anything, you're best bet might be to do it up North. If you do decide to do it in Ireland though, you'll only have to sit four ground exams as opposed to seven in the UK. Whatever you decide though, make sure you talk to both the IAA and CAA first to get the official word before you decide anything. A few schools in America were trying to tell me that I could come over and I wouldn't have to re-sit any ground exams!
Best of luck,
Sprawler