I've had the same thing with a different tuner (a Fender, I think): I made completely sure that I had it turned off, even turned it off and put it in a drawer to make sure that I would not accidentally turn it on - and a day or so later, the battery would be empty.
I don't think it has anything to do with the brand of tuner, just an individual faulty one (a short somewhere, maybe?).
None of my Snarks (and I have about half a dozen now) have had the same issue; over the course of a year or more the display slowly gets dimmer, but that seems to be expected.
Especially because I start them out with the batteries that they come with, which are probably not the freshest / best quality to begin with.
Whereas the one micro headstock tuner that I bought came with a battery that was drained within half an hour of installing it - but the new battery I put in is lasting fine (I don't like those as much as I like the Snarks, though; they seem less accurate and less sensitive and the pulsating metronome - which cannot be switched off without switching the entire tuner off - gets on my nerves).
I don't suppose you still have a warranty on your Snark?
Eta: I love your taste in Ukuleles, by the way!
I have two Bruekos myself, and am madly in love with my new Mainland Mango Concert at the moment.