I started doing some research for a similar trip about 9 months ago.
One of my ideas was to hire a bike in Ireland but also spend a few days on the Isle of Man (after all the TT ruckus had died down; late June).
I stumbled on a new and used bike dealer in Belfast called Phillip McCallen Motorcycles. (I know it's in West Belfast, but they do offer to pick you up from the airport)
After an email enquiry I was contacted by a very accommodating bloke called Sam McClurg who answered all my questions, then politely kept in contact over the past few months.
I have had to put the trip off to 2014, but I am confindent that these people will live up to expectations.
They insure the bike for Eire and Northern Ireland, Isle of Man and mainland UK, so you can go where you want. I was also thinking the Yourshire Dales and Lake District might we worth a look. You have to pay the ferry fares. Look for Isle of Man Steam Packet Company
They are Triumph, Kwakka and KTM agents, but recommended the Triumph 800 Tiger as being most suitable for Irish roads, and they will supply them with luggage.
I seem to remember that they can hire riding and wet weather gear as well.
I haven't got the emails to hand, but mey recollection was that the hire was about 385 GBP per week which is about $600 US (but don't quote me on this; whatever it was it didn't sound excessive).
One of the reasons to consider a side trip to the Isle of Man is that after TT fortnight, the open roads again become an open speed limit; how cool is that?
Tim in Australia