May Driving Itinerary

In general it looks like a very well planned itinerary. Just a few suggestions:
If you're arriving from the US after a long overnight flight you might consider just taking the train from Dublin to Galway and picking up your car there, if only to avoid having to hit the road immediately after arrival when you'll likely be pretty fatigued. Explore Galway on day one, get a good night's sleep, and start driving after you're both more refreshed.
Although we love Clifden, basing yourselves in Galway would be more central for day trips both north and south.
The coastal drive from Galway south towards Dingle is quite pretty - lots of pretty seascapes along the way, and the pretty villages of Kilkee and Kilrush are good stops for lunch. There's a nice cliff walk in Kilkee that's a good place to stretch your legs. Pretty views too. Then the Tarbert ferry across the Shannon would get you on your way to Kerry. Not the faster route but certainly more scenic than the motorway south.
Personally I'd suggest staying near, but not necessarily in, Killarney. Even in May you'll be dealing with the crowds, and the traffic congestion getting in and out of town can be a real bear. Kenmare would be a good alternative. You could also consider staying somewhere between Castlemaine and Glenbeigh to be centrally located for exploring the ROK, the Gap of Dunloe, the Skellig Ring, and Dingle.
We prefer to stay near the airport the night before we fly out, and I like to have turned the car in too in order to minimize the stress of the last day ... but that's just me. To each their own. There's nothing wrong with doing it as you propose if you get an early start and plan for unexpected delays.
It'll be great.

I just got done doing all the driving for 12 days in Scotland and I am currently in Ireland also doing all the driving. It has taken longer to get everywhere than I expected and one does want some days with no or minimal driving. I say that even though I don’t think the driving in Scotland or Ireland is difficult.
Also, you don’t mention it, but if you love Irish trad music like I do, Dingle is a great place to hear it! Went to a fantastic concert at the Dingle Music Shop the other night!❤️

Thanks, Robert and diveloonie, for your feedback!

After going through these ideas and several more hours of research, we settled on the following itinerary. We picked up the train to Galway suggestion and left the rental car to handle at Sunday lunch. I elected to add in an overnight halfway to Kenmare just to give us flexibility to go slow that day and the day after. I think I might splurge on a castle stay here.

We also selected Kenmare because of the great restaurant reviews and discovery of some information about the Ring of Beara. Finally, we will overnight at a Dublin airport hotel after dropping off the car. We will run into Dublin if we have the energy but I bet we just want to take an extra nap by that point!

Saturday - Arrive, train to Galway. Overnight in Galway.

Sunday - Galway/Clifden/Sky Drive/Connemara. Pick up rental car this day (lunchtime). Overnight in Galway.

Monday - Galway/Clifden/Sky Drive/Connemara. Overnight in Galway.

Tuesday - Cliffs of Moher, The Burren. Overnight north of Limerick.

Wednesday - Castles on the way to Kenmare (or something further afield). Overnight in Kenmare

Thursday - RoK/Skellig Ring/Ring of Beara. Overnight in Kenmare.

Friday - RoK/Skellig Ring/Ring of Beara. Overnight in Kenmare.

Saturday - Head toward Dublin via Rock of Cashel and Cahir. Drop off car. - Overnight at Dublin Airport

Good moving choosing Kenmare.
Beara is indeed a pleasant alternative to the ROK. If you want to make a meandering day of it you could drive out to the end of the peninsula to the only cable car in Ireland over to Dursey Island - capacity: 3 people and a cow, which will give you some idea of what you'll be in for. West Cork was extremely hard hit by the famine years and Dursey was essentially deserted, so there's not much there today but the ruins of several stone cottages, an old church, and grazing sheep and cattle. It's a very evocative place. The word has gotten out about it, however, and on our last visit there was a line (and a long wait) to cross so we gave it a miss and just hiked around the head of the peninsula - still a very pretty hike.
If you're comfortable driving on the left, and on narrow rural roads, you could drive up and over Healey pass either going into or returning from the village of Adrigole. It's not for the faint-hearted though - it gets narrow and twisty turny at the top. Beautiful though, with many spectacular vistas as you descend.
And if you really, really want to get off the beaten track there's an abandoned famine village tucked away in a remote valley that requires about a 2 mile hike up and over a gentle ridge to access it. It was home to about 30 families back in the early 1800's but was abandoned around 1850. For my wife and I it was like hiking into Brigadoon - liked it so much we went back twice. Let me know if you're interested and I'll try to dig out the directions.
Safe travels.