
I don't think any of the domestic airlines will carry animals.
Really, the ferry is probably your safest bet. The fast crossing between Stranrear and Belfast takes less than 2 hours. I'm in NI and I know lots of people who have travelled from England with pups this way and it has been fine. Pup stays in the car (crated, preferably) and will likely sleep through the crossing anyway.
*edited* ah, just seen klb's response. Yes, Dublin may be quicker depending on journey time to Holyhead vs Stranrear. It's motorway pretty much all the way to Belfast from Dublin and can be done in 90 mins - 2 hrs depending on traffic.