You won't be allowed to bring your dogs to any beaches but the dog beaches during the day, just so you know. If you bring your dog to a non dog beach they will ask you to leave, the lifeguards usually just ask you to go but I've seen people get ticketed by patrol officers.

Your budget is fine but finding a place that allows dogs that large in any of the beach areas is going to be difficult, as stir_friday already said. Places with space that large are usually going to be houses and finding vacancy in the beach areas with that much space is going to be really really difficult, the vacancy rate in OB where I live is incredibly low. I don't know of any apartment complexes around me in OB that allow dogs over 25 pounds. I would look for a rental house for the beach areas you mentioned, not apartments.

You will have more luck in the North Park and surrounding areas but that's not "close" to the beach, even though its only a 15 minute drive.

I don't want to be a dick but your stipulations about your dogs and wanting to be a mile from the beach are going to make your search, really tough even though you have a budget that can be worked with. If you didn't have large pups you would have no trouble in the beach areas on that budget.

On the plus side, San Diego is an incredibly dog friendly city basically everywhere, I've never seen a town with so many animal friendly restaurants, so if you don't mind not being "near" the beach, you will probably be fine. North Park and the surrounding areas are still close to downtown and will have more space options for you and your pups. It is easier to find rental houses in the North Park/Normal Heights areas that have patios and things like that.

Good luck.