Flame's right, dogs are allowed on the beach at Fisherman's Walk down to Southbourne and then there are restrictions again. They aren't allowed on the beach in the other areas between 1st May and 30th Sept, but they are allowed on the prom on a lead.
You could go to Boscombe where there's a car park just above the beach, take the dog onto the prom and tie him to the railings, then picnic on the beach just by him.
Yes, it's a huge long beach with a promenade behind it. You would be able to walk either way on the prom with him. There are signs up on the prom to tell you where you can and can't go with him, or if you ask in the Beach Office or one of the lifeguard huts they will be able to give you a leaflet.
Have a lovely time. I'm as I have to work tomorrow, but I gather the weather is improving so I shall be down there Saturday and Sunday. I'm going to make the most of it as it'll be Autumn before we know it!!