It's very common for non-UK residents to set up a UK Ltd company, most of the time due to the fact the there is no set minimum amount of value of the shares that are needed for the incorporation (e.g. £1 is sufficient and you are still not personally liable. This company can be run in any EC country).
If your Irish company is registered for VAT in Ireland you can reclaim VAT on training made in the UK. Have a look here.
http://customs.hmrc.gov.uk/channelsP...E_PROD1_026296
I believe a third alternative is that you register a company in Ireland and then register for VAT in the UK. However, I believe the alternative as of above is the best one.