I did a similar trip a few months ago at the beginning of June and like @Tomkat above I was solo on a KTM 790 Adventure. My biking friends were off to France but I was saving my 90 in 180 days for right now, so I needed to be somewhere not in the Schengen zone.
I decided to concentrate on a relatively small area of the Wild Atlantic Way in the south west of the country. Ireland is not cheap for accommodation and I chose to stop overnight in hostels. I would encourage you to do the same as (a) it's a long time dark in winter to be sitting in a tent, and (b) you waste an awful lot of time each morning breaking camp, especially if wet.
Day 1: I woke in London at 4:30am, started out at 6am, and took the 1pm ferry from Fishguard to Rosslare, arriving 4:30pm. I slept in a day cabin during crossing and overnighted in Sheila's Hostel in Cork (fine hostel).
Day 2: The next day I rode the Ring of Beara (green route), then overnighted in the Railway Hostel Killarney. The Railway was a major disappointment, so I can't recommend that one.
Day 3: Rode the Ring of Kerry (blue route).
Day 4: Rode the Dingle Peninsula (red route and my favourite of the three) and overnighted in the Grapevine Hostel in Dingle (lovely hostel).
Day 5: Rode to Dublin, overnighting in Jacobs Inn Hostel (real upmarket)
Day 6: day in Dublin
Day 7: Ferry to Holyhead, rode home.
My trip was close to midsummer, so long daylight hours and great weather. In the depths of winter you might struggle with just four days, five would be better if you can manage it, and even then you would need to knock out one of the 'rings' and the day in Dublin. In fact if you are leaving from Rosslare, it doesn't make sense to go north first to Dublin.
GPX file is attached with routes and 40+ waypoints (I had to remove the tracks to and from the UK ports to comply with file size restrictions).
The Irish Road Trip Guide is a useful planning site. I booked hostels through
HostelWorld. Happy to answer any questions.