Best time of year for 15-16 day Ireland trip

We spent 2 weeks in Ireland in May and the weather was fine most of the time, on a couple of days the wind made the temperature seems chilly but the rest of the time is was perfect. We didn't really have that much rain, either.

Either May or September would be good months to visit.
Don't know if either technically qualifies as the shoulder season for airfare or accommodations, but I can tell you from some experience that the crowds are down and the weather is good either month. Would expect that availability would be better and that prices would be a little better during either month.
If you're into gardens I'd recommend May when the countryside is in full bloom.

Fares to many countries are less expensive in the off season, generally defined as before June and after mid-September. However, over the years we've always found cheap airfares ($500-$800) to Ireland for July-September travel. Just do your research early and book directly with the carrier in case complications arise.

A synonym for "weather" in Ireland is "rain." No matter where you go in the country expect it nearly every day. Usually it's light and intermittent, but sometimes lasts all day. The Western (far more scenic and culturally rich) counties, especially Kerry, get a lot more rain and are cooler in the summer than those in the East (50-60 F vs. 60-70 F). Rick's Ireland guidebook shows an average monthly difference in days without rain for Dublin as four or less. I'd double that insignificant differential for the West.

Therefore, I'd base your travel dates more on what you want to see and do rather than the weather. Uh, I mean the rain.

I spent a week in Clare in early April 2019 and we had beautiful weather. Not once did it rain.

We were there one July and it rained every day but the day we were at the Dingle Peninsula. And cold.

Whenever you go you'll want a raincoat and at least a light sweater/jacket. Layers will be important. Hotels/B&Bs will likely be less expensive in the shoulder season the exception being Dublin which is always expensive, especially during the work week.

As a general rule, travel to Europe is more expensive June-August, including airfare and lodging. We normally travel April-May or September-October since prices are cheaper and the weather and temperatures are still ok. Our Ireland trip was in October and we need jackets in the evening and on those cloudy, windy days.

We've been to Ireland twice in May and had hot sunny, weather both times. Never used my raincoat. I only prefer June trips because the trad music is more accessible.