eh no worries, its a lot less convoluted than I made it sound, ....I think?
I have tried equip on the lantern, it is not tool number 4 - but anyway. You start with a lantern above the crafting bench
its pretty easy to just chuck it into your mining crate and close the lid - I don't think carrying the crate really slows you down, but if you feel it does, you can just lob the crate towards the cave (bonus: the lantern inside helps (potentially hurts) your throw, just have to throw it right)
Then just load up with whatever you can mine.
don't worry about "losing" space for ore
you can actually stack the ore higher now thanks to the lid, which will close through objects, locking them inside - so now the stack can be higher and you won't have to worry about it toppling over. (though maybe, some of "you" are daredevils and stacked it absurdly high anyway...)
Having the Lantern as tool 4 would be the only thing easier and better. (since you have your pick/hammer still out while carrying the crate, the light would still be out, and work far better than when its in the crate)
But, for now, having it in the crate is 100x better than just having a lantern. (which we do have "just a lantern".. its above the crafting bench) because you can only carry one thing at a time... and the way I speak of gives you both light enough to see what you're doing and where you're going, and carrying both things
if you couldn't put it in the crate, you'd have to choose between carrying the lamp, or carrying your ore out of the cave, and that would be the real tedious thing.