Kia Orana, How old are the kids? Some places will take kids over 12. Your requirements limit your choices. Nothing here like a "pension". The only places that include breakfast and dinners would be the resorts which you have ruled out.
The Paradise Cove has nice family units and they will provide breakfast for a price. e-mail firstname.lastname@example.org website http://www.paradisecove.co.ck/
Right next door is Puffy's which is open for evening meals except Sundays. It's primarily a fish and chips place but they do have other items on the menu.
You might also look at Popoara Ocean breeze villas. They offer packages with breakfast included - I suspect at the Koru Cafe which is near by. They also have a restaurant so they might be willing to work something out for dinner as well. They are not on the best swimming beach but there is one very close by within walking distance. E-mail email@example.com
If you haven't looked at this website, check it out aitutaki-accommodations.com/Aitutaki_Accommo…
Good luck with your search
Edited: 1:47 pm, February 19, 2013