Or you could let the reeds dry out between sessions. They'll last just as long, and you dont have to slosh a jar with liquid around and make sure they dont break in there too.
Btw, you should change the water (if you're gonna do the pickle jar thing) far more frequently than once a month - eeeeew.
Yes, it seems to me at least that the warping that occurs when the reed dries does not matter if it's soaked sufficiently before playing. Mine are usually like fried bacon when dry, but when soaked they get perfectly straight again. Of course the soaking time needed can be a problem if you're in a hurry, I find it takes as long as 10 minutes, 5 minutes being a normal average, to get the reed straight if it's really dry, harder reeds seem to take longer than softer ones which makes sense as they are thicker.
I frequently soak mine at home before going to a lesson etc. and still they dry out a little during the travel, so I have to keep them in my mouth for a while before they start working properly.
Haven't tried keeping them wet all the time, so I don't know if they'd last longer that way.