There are two ways to approach short trips: the cram as much as possible in approach, or the approach that accepts you'll not see more than a few places but resolve to really enjoy those. You are taking the second one and I think that's a really wise choice.
That said, when you have so little time it is important that you use it well. In your case I think the whole thing pivots around the transfer to Dublin and how to fit that into a good itinerary. Assuming you start in Dingle, that journey is realistically 4hrs 30mins with no stops along the way, and for the most part a bit boring. So I've some suggestions about that below.
I am recommending you do Dingle rather than the Ring of Kerry partly for this reason - less driving, more time out of the car. Other reasons are that while everywhere in Kerry is busy this time of year, Dingle is the less crowded option. The weather is a consideration too and, sadly, it likely to be cloudy while you are in Kerry and the Ring depends more for its magic on the amazing sea views. On a cloudy day you often just see mist, no views!
One more thing. The Rose of Tralee festival will be in full swing on the Sunday you are in Kerry - there will be more people than usual about and Tralee especially will be, frankly, chaos. So I'd not linger or even stop there.
I think you'll do the Burren in a good day, provided you stay locally too. Plan to spend the next day exploring Loop Head - nothing like as busy as any of the other places you'll go to and really an overlooked part of Ireland. Try to fit in a visit to the absolutely wonderful Scattery Island.
Then Kerry the next day.
If you get as early a ferry as possible to Tarbert, you'll be in Dingle in an hour and a half and can spend most of the day exploring the peninsula (you could happily spend the entire day in Dingle itself, but don't!). In the early evening, leave and make a bit of headway on the Dublin journey by driving to Adare and staying there overnight.
Now, there are two ways to do your last full day. Either get to Dublin as quick as you can, to maximize the time in the city, or meander your way to Dublin with lots of stops and get there in time to spend an evening in the city. Really it's your choice, there is no 'best'.
If you choose to go direct to Dublin, at least by staying in Adare you'll be able to make the most of your time as you'll be there in not much over 2 hours.
If you go for the meander, I'd take the route on this map, and stop in Cahir, Cashel (I can't believe I've no article about the Rock of Cashel!), Kilkenny and possibly Kildare (for the National Stud) if you've time and it isn't raining.
On accommodation, recommendations may be kind of pointless because with such short dates you are realistically restricted to what's available. That said, the Adare Village Inn is a very good, central and reasonably priced place to stay in Adare. You'll need to call or email them to check availability, as they are not on booking sites.
If you are planning on splurging on one really really good dinner during your stay, this isn't a bad night to do it. My favourite restaurant in Ireland is just a short distance from Adare: The Mustard Seed.
I just checked. The Mustard Seed have a special deal of Dinner, Bed and Breakfast available on the night you'll be in Adare for €129 per person (or €115 if you stay in a smaller room). It's not rock bottom pricing, but given that their normal rate for room only would be €130+ and dinner would cost €60 per head it's a really fantastic deal and if you want to treat yourself you honestly couldn't do it in a better place. I'm actually so excited about discovering this deal that I'm going to avail of it myself in the near future!
I hope you have a wonderful time, let me know how it goes!