July in Ireland

The average high temp for Ireland in July is 68 F with 2-3 days of rain. You are more likely to be cooler than hot. Wear cotton nightwear, if you sleep hot. I usually wear a Tee shirt dress that's long enough to cover up, in case the WC is in the hall. The main reason they don't have AC is that you might be able to open a window to catch a breeze.

They don't have A/C for the same reason they don't have lawn sprinklers. They don't need them.
If you're worried about it, look for accommodations that have A/C. Booking.com and other similar sites offer A/C as a search filter. Book directly with independent hotels, of course.
I'm not much for hotels, but I don't recall the duvet-only being a problem in Ireland. And the duvet can come out of the cover or you can ask for a sheet if you're in a hotel. Outside of the concrete jungle of Dublin and especially on the coast, it's likely to cool off considerably at night, even if it's hot during the day. Which it's usually not.
Last July, I wore my souvenir wool hat one day, but in 2018, it was quite warm and hadn't rained in a month. I don't remember ever being uncomfortably hot at night, but we did not visit Dublin on that trip.

I visit Ireland regularly and I've never been warm enough to need air conditioning. As others have said, even on warmer days in the summer it does cool off quite a bit at night. It would be very unusual if you needed it. In fact you may even want to bring a jacket.

On our most recent trip we stayed in an older Dublin hotel that was built of brick. Though daily high temps never exceeded 75F we found that the place absorbed what heat there was during the day and was slow to cool off enough in the evening for my wife who is sensitive to such things.
Since then I've been under direct orders that on future trips (while in the city anyway) we'll be looking for a place with A/C ... which indeed can be hard to find.
We've never had any issues when staying in more rural areas - only in Dublin city itself.

You might consider packing a small portable fan - there are many USB powered options (see these for example). I wouldn't count on a hotel in Ireland having a fan to borrow as you might be able to do in much warmer countries. You could also pack your own top sheet in a material that suits your sleeping style, just remember to take it with you when you change hotels.

On our trip to Ireland I did find that the bedding was heavy. The weather wasn't hot, but sleeping under a thick duvet sometimes got stuffy. So when I needed to I would pull the duvet out of the duvet cover and sleep with just the cover as a top sheet.

We were in Ireland, two years ago in July/August. I wore my lightweight Merino wool sweaters most days and a nano puff jacket in the evenings. Of course, I understand they occasionally have a hot spell so if you’re concerned you could always pack a lightweight sheet with you.