I've never tried it, but I wonder if it might be possible to reflow QFN with a hotplate or electric skillet. It also might be possible to use a QFN as a dead bug. Finding some solutiuon may be important because I think you'll see a lot of QFN-only parts in your search. There are QFN-to-DIP adapter boards available for $15 for 10 of them. If their pads extend bit, there may be a way to solder them.
If you could use two 5V solar panels instead of the 8V, it might be possible to pick a processor that works anywhere between 2.4V and 5.5V, such as the Atmel AVR parts, and then you wouldn't need a regulator. The Atmega328P, for example, can do that at 8MHz I believe, and goes into deep sleep at something like 100nA. And it even comes in a DIP option.
The battery itself will probably have a protection circuit built in, which typically shuts it down at 2.4V. The protection circuit may draw more current than a sleeping MCU, but it's still only a uA or two. The protection chip is the DW01.
If you can solve the QFN problem, you might look at the MCP73871, which is a couple bucks. I think Adafruit has a solar charger based on that chip, and there are cheap charger modules using it on Ebay and Aliexpress. Using a module might get around the QFN problem. This particular chip has load sharing built in, which you would otherwise have to add.
I'm told that having someone like JLCPCB do the entire build is now actually within reach cost-wise so long as you use their parts. But I've never priced anything out.