I stayed in Doolin once. I also visited Aran and the Cliffs of Moher each twice. (Sadly, the Cliffs of Moher now charges a fee; when I first visited the cliffs in 1993, it was free.)
The trip to Aran would be nice because of the beauty, wind and antiquities of the islands. I do not know if bicycle rental agencies offer special seats for an 8 month old, but you could email the company I used, Aran Bike Hire: https://aranislandsbikehire.com/. There are other companies, too, if Aran Bike doesn't have seats for kids.
Since you'll have a car, maybe you could go to a sheep petting zoo because kids love animals. There's one near Westport, Ireland, which is north of Doolin about 60-70 miles. The farm is Glen Keen Farm: https://glenkeenfarm.com/. The petting farm is actually in Louisburg, Mayo. I've been in Louisburg. You'll practically be in the shadow of Croagh Patrick (https://www.mayo.ie/attractions/croagh-patrick), which is sometimes called the holy mountain of Ireland.
You might hear Irish spoken as the day-to-day language in Doolin. You'll definitely hear it on Aran if you go.