First things first, its not exempt its zero rated. Once you give your Irish VAT number, they should zero rate the supply.
For the accommadation, meals, etc, you can reclaim the VAT from Irish Revenue (who in turn get it from the UK Revenue).
'Under EU Directive 2008/9/EC (effective from 1 Jan 2010), you can claim all EU VAT incurred through the Irish Revenue, once you are registered for VAT in Ireland'.
You can read more here
http://www.revenue.ie/en/tax/vat/refunds/unregistered-persons-vat-reclaims-faqs.html (scroll down for Registered traders).