One town for 4-5 days in West/Southwest Ireland

Will you have a car?

I knew I left something out! Yes, we will be renting a car.

We don't do "relaxed", but I think I'd pick Dingle. Crowds won't be a problem in April at any of your picks, although music might be scarcer in Kenmare and Killarney. Based on our visits, we like Galway, Dingle, Kenmare, Killarney in that order. Doolin would be tied with Dingle. We're always there in May.

From Dingle, you can visit Blaskett (include great museum), lots of nearby old ruins, Anascaul Lake hike, and of course back to Killarney NP, Ring of Beara, Ring of Kerry, maybe even Skellig!

Though most here (ourselves included) usually advise steering clear of Killarney, in your particular case it might be your best option given your interests. Lots of restaurants, pubs, and shopping opportunities plus it's the starting point for most of the local tours if you're interested in something like that. Killarney National Park has some very pretty walks, and Muckross House is a favorite of many here. In April you'll be ahead of the tourist crush of summer ... which starts around the end of May.
We usually stay outside the town - somewhere along the corridor between Cahirsiveen and Castlemaine - in order to be centrally located for easy day trips by car to the Dingle peninsula, the Ring of Kerry, Valencia Island (a favorite), the exquisite Gap of Dunloe, and back into Killarney to visit the town and NP without staying there.
With your early arrival in Dublin you might consider just staying at the airport and catching the direct RyanAir morning flight to the Kerry airport near Killarney - which takes about an hour. You'll likely be running on adrenaline anyway, and you won't be able to check into your Dublin accommodation that early unless you've booked the night before, so you might as well use the time to continue on to Killarney , spend your time there, and save Dublin for the end of your trip ... again via RyanAir.
There are rental car outlets at the Kerry airport, FYI.

I'd go with either Galway or Dingle, both more interesting than Kenmare in my experience. Lots of restaurants, B&Bs, pubs and trad sessions (prehaps held less often in the spring than in summer) in both. Killarney, with all the buses coming and going, is indeed too touristy, although there are some beautiful places to visit nearby.

From the Galway city dock you can take the ferry to one of the three Aran Islands for either a day trip or an overnight or drive 24 miles west to Rossaveel for the same.

From the downtown Dingle pier, hop the Peig Sayers for a day-long visit (or now, an overnight in one of the island's cottages) to Great Blasket Island. If you're on Blasket on the right day, you be able to enjoy a fascinating re-telling of the island's history and its inhabitants presented by a bi-lingual historian. If you have a car, then a drive around Slea Head Loop is another option.

https://www.aranislandferries.com/#indseo

https://www.greatblasketisland.net/

Thanks for the replies so far -- this is great information!

For a bit more detail, we plan to save Dublin for the end of the trip, but we're fine spending the first night there if it makes more sense than trying to do the long drive after taking an overnight flight. I didn't think about just taking a flight from Dublin to Kerry, so we'll definitely look into it that option. Has anyone here done the full drive across from Dublin on that first-day adrenaline, and if so, how did you make out?

I think a lot of what we want to do is in/around the Dingle Peninsula and ROK, so we'll probably eliminate Galway. So I suppose it's down to Dingle, Killarney, and Kenmare.

Driving across the island to either Killarney (4.5 hrs) or Dingle (5+ hrs) would be a bad idea I think. Once that adrenaline wears off you'll be asleep at the wheel - doubly hazardous if you're not familiar with driving on the left, the local rules of the road, etc. You'll want to be especially alert generally have your wits about you before setting off in a car ... especially on your first day.
The short drive from Kerry airport into Killarney or Dingle shouldn't be a problem, but anything longer than that would be risky.

With a car, you can stay anywhere. With teens, you'll want self-catering (IMO). My suggestion is to use your preferred booking site (mine is AirBnB, which I know a lot folks don't like, whatever), choose your parameters (cost, number of BRs and BAs, free on-site parking, full kitchen, whatever you need), and zoom out on the map to encompass all of the areas you're considering. Then see what's on offer. You might well find something with a sea view on the edge of a village with pubs, etc., whatever suits your taste.
In addition to flying to Kerry Airport, there are buses and trains that can get you there from Dublin and car rental agencies on arrival. I would suggest broadening your search to include West Cork (Kinsale, etc). You can't go far wrong in any of the areas mentioned.

Based on your description I would choose Dingle. Highly recommend lodging at Greenmount House.

For a first night stay in Dublin would recommend Ashling Hotel - super breakfast buffet.

I've tried the marathon drive after long-haul (4am out of the house, 1pm departure from Seattle, noon arrival in Dublin) and I don't recommend it, especially if you're new to left side driving. We drove from Dublin to Kilkenny to Clonakilty and I was exhausted. I'm still game for a drive to Kilkenny or Belfast, but not much farther.

If you're only coming from the East Coast or can get some sleep in flight, it's not suicidal.

Don't overlook the car drop-off fee to use the RyanAir idea. There's also train connections to consider, although Ryan is much faster (but less scenic).

One other point to consider is if you want to have a beer while listening to music, you probably shouldn't arrange lodging that requires a drive, unless you get a taxi. I have a very functional liver but I never drive after a pint or dram in Ireland. Also, if you're "in town" you won't have to always stick together if some people want to have a little time to shop/mingle with the local kids, etc. without driving constantly.

For info, there is no drop fee if you collect and return your car at the Kerry airport. Per AutoEurope the one-way drop fee between Dublin and Kerry is about USD $82, so a return drive back across the island would still be affordable if you choose to do so.
In case you're not already aware, one important detail to consider when shopping for a rental car in Ireland is the matter of the insurance cover. Most regular US credit cards specifically exclude coverage, though many of the premium cards do. You'll want to have that detail sorted out well in advance in order to avoid any drama at the rental car desk upon arrival. We usually opt for the zero deductible coverage thru AutoEurope, though others here prefer to trust their premier cards. To each their own.

There's also train connections to consider, although Ryan is much faster (but less scenic).

If you've landed very early (presumably around 6am) then the Ryanair flight leaves at 9am. By the time you've hung around for that flight you can be most of the way to Killarney or wherever down in the South/South West on the train, so speed is illusory; and have a nap on the way. And if the flight is late and you miss the morning flight you are buying a new Ryanair ticket (if available) or hanging round all day. On the Train or Bus you just turn up and buy a ticket, and away you go. The train to Killarney/Tralee is every two hours, and to Galway more often than that.
The bus to Galway is at least hourly.