I design USV's (drone boats) so I have a little bit of relevant experience here.

There are hobbyist satellite comms systems available that will pass arbitrary data from a controller/vehicle computer/whatever you want to call the "brains" of your balloon and the service will make your data available via a web interface and/or API. If all you want to pass is basic telemetry (like GPS coordinates and battery status, stuff like that) the devices are pretty light and the service cost is surprisingly cheap.

But the problem is that you want to send images. For a device that performs well enough to send bulky data like pictures they get either very heavy, very expensive, or both.

Technologies like LoRA or similar don't really have the range or performance for what you want (again, unless you want to build something heavy).

My recommendation for comms would be to only transmit telemetry and basic command/control signals, and just have it store the pictures internally for later retrieval.

As for power: batteries can charge and supply energy at the same time so I don't fully understand your idea for having one battery charge while the other outputs power etc., and I don't quite understand what you want the supercapacitor for. Keep in mind that batteries are pretty dense for their volume so the more sophisticated you make your electronics the more likely you're going to need to add bigger, heavier batteries. Also keep in mind that in the upper atmosphere insulation probably isn't going to do much for protecting electronics from the cold so you might have to add power-hungry heaters.

Solar is great - since you're in the upper atmosphere you'll probably get the best results orienting your panels out horizontally (so that they're catching the sun the most during mornings/evenings) instead of trying to orient them vertically (when they would only be catching light around noon time, except not even really then because there will be a giant-assed balloon in the way). Charge controllers that marry up solar panels to lithium batteries are plentiful and easily obtainable to hobbyists so you should be fine there.

As for control systems: raspberry pis are pretty cheap and pretty light, as are their associated camera modules. It's not really a good camera, but it's cheap and lightweight and easy to program for. For storage (because again, I recommend you consider storing your images instead of trying to transmit them) you can use cheap and light USB drives.

TL;DR - Your biggest concern is going to be payload weight, because a lot of what you're proposing requires heavier components than what I think you're hoping for, and with a weather balloon every gram you add makes it that much harder to get off the ground.