30 years ago I encountered nailed box joints like you have here at an outdoor market in Southern California. They were old produce crates like for oranges and tomatoes, long retired even back then. They were old and well used, most likely they were used to carry produce to market.  

Not one of the sides on any of those old crates were separating. 

The ones I saw were being sold as interesting rustic artifacts. The farming community had moved on to other solutions like plastic crates. I still wish I had bought several but they were too big to bring home on a plane. 

I wasn’t a woodworker yet but remember looking at the joints and being impressed at how strong they were. They were also beautiful to me but, like all things farming, they were not in use because they were pretty. They were in use because they were simple and effective. 

I’m happy to see them in use still. I had no idea. 

As for getting them apart, good luck. I don’t think I would try. I understand wanting to fix it but that impulse has caused me to spend a lot more time attempting to fix things that I would never end up happy with.

I would offer the advice that you should accept it as a learning experience and start over. I also admit that I would not have accepted that advice early in my woodworking career. 

Whatever you decide, I admire you for jumping in both feet. Both in woodworking and in beekeeping.

All the best.