Could try Sandsend or Robin Hoods Bay near Whitby. Dogs are allowed on both beaches all year but they aren't allowed on the beach at Whitby in the summer. There are alot of good campsites in both area.
High Straggleton Farm at Sandsend does allow dogs but it doesn't state in their listing that they do.
it be right on your limit distance wise, but penryn bay, angelsey is a great site and just 5 mins walk to a huge sandy beach wich is excellent for dogs and people alike !
I would second Esther's suggestion - Freshwater East is within walkingh distance, but if you are pitched at the far end of the site, then it is a bit of a hike.
Have a look at North Norfolk. We have had our main holiday in that area for the last 2 years as well as a couple of short breaks. There are some limitations to dogs on beaches but not too many. It is a really nice area.