Gnobak, as someone with a long-term eye issue, the private options for eye care are quite limited The Eye and Ear is the only specialist hospital in the country and it is a public hospital only (although they will happily bill your private health insurance and put you in a private room if you ask). The hospital is not equipped to do anything beyond that for private patients - you need your GP to refer you to a private consultant when you get out of hospital if you want to see a private consultant going forward. There are loads of eye specialists in the private system and you can be looked after well within in.
If you have an eye emergency and are sent to the A&E, neither Blackrock nor Beacon private A&E are set up to take eye emergencies (don't know about Mater private) so eye emergencies in Ireland are invariably seen via Eye and Ear Hospital, landing you back where you started. The private hospitals have loads of facilities for eye surgery and medicine - they just don't have emergency access to them
As the posters above note, you have 2 choices - one is to continue with the Eye and Ear, which has the most comprehensive care in the state, but is a bit of a pain. The other is to ask your GP for a private referral to an eye specialist and they will see you. Make sure you get the name of the specialist and if you don't hear in a few weeks, call the secretary of the consultant directly to follow up.
If you take option 2, you need to do that before there is an emergency - as basically all urgent eye care is referred to the Eye and Ear public hospital, an eye emergency will invariably land you back there.