Oh, dear. I am sorely tempted to suggest you go on eBay and find a Yashica T4 Super D (usual selling price is around $150). This is a truly marvelous camera, with a fantastic Zeiss lens, easy to use, tough, and water-resistant. And its C123 battery will last for days. Alas, it is a film camera. But of course, you could shoot Kodacolor 100 and then have your camera shop make digital copies of each picture, which costs next to nothing. I swear, youd really like this camera. I have one and will never give it up. It takes incredible pictures.
But, I know, film is so...well, 1990s. This is a digital age. Second choice: A Canon G11 ($499). A pro-level digital camera, with a fantastic lens, 12-megapixel resolution, great processor, LCD screen, and a viewfinder. It has a Canon proprietary battery, but you could charge it with a reasonable solar charger such as Brunton Solaris 12 ($410), a very lightweight (11 ounces!), flexible solar charger.
No, the G11 is not waterproofyoud need to add a protective case. Canon makes a true underwater case for $200 (the WP-DC34). Or, get an Aquapac underwater case ($40). Its a flexible housing with an optical-quality lens cover thats waterproof to 30 feet or so. A good deal for nearly any camera.
Or, get an Olympus Stylus 8000 ($380). Not the camera the G11 is, but pretty good (12-megapixal, 3.5x zoom, viewfinder, and LCD screen). And its waterproof to 33 feet out of the box, as well as capable of withstanding a six-foot drop. Darned tough.
Meh, but again, a proprietary battery. But here once more you can find a decent, light charger and make it work. Better cameras that take AA batteries are rare, I fear. And nothing that meets your specs, really.
I go back to the Yashica. Buy it and 12 rolls of film, and you are set.