We have just come back from Pefkos, Rhodes, a fortnight same dates as you require. The day temp was mid 20's and a bit cool in the evening the first week, then much hotter and able to sit out with no cardi at night second week.
We stayed at the Island Blue hotel, part of a group including the Pefki Islands and the Lindia Thalassa. The accommodation is mostly self catering, but you can have B AND B at the Island Blue. The breakfasts are very good, the dinners less so sadly. The advantage is that you can use all the pools, bars, restaurant etc in the complex. The L.T. has a spa, and there is a supermarket close by. Beach is 200m away down stone steps. It isn't very deep from land to waters edge, but on the plus side you are close to DC playing in the water, which is shallow for quite a way out. The hotel pool is good with a shallow 40cm deep section for toddlers. If you are lucky, you can watch weddings happening at the little chapel.
It isn't far to shops and restaurants, but a bit hilly. If you can get to ones up the hill, you are rewarded by spectacular sunsets.
There are some two bed apartments, which we had although we didn't need! One self contained bedroom with en suite shower room, and a main bedroom with kitchenette and shower room and a balcony with table and four chairs. There is a fridge, kettle, toaster, sandwich toaster, iron and ironing board and all the usual pans, glasses, crockery and cutlery. Hair dryers in shower rooms, TV in main room. I know you don't want to self cater but you might want a drink when the DC are asleep. Cots seem to be the travel cot sort.
Culture: Lindos is a short taxi ride away (8 euro one way) you could get the bus but they are often full and you don't want to climb up to the acropolis midday. The shops are quite quaint, but it's not easy with a buggy as it's on a hillside and there are a lot of steps.
You can fly from Manchester on Weds and Sats, 4hrs 15mins flight. We went with Thomsons. We paid just under 2200.00 for the two of us B and B. The official rating is 3* but I would rate it a bit higher.
If you're interested, book early as there are only 34 apartments in addition to the 52 rooms and they get booked up by wedding parties quickly.