You could put the dry ice into pure ethanol (everclear maybe?) which would create an ethanol/dry ice bath at -72 degrees Celsius. Of course, this is much warmer than liquid nitrogen which is ca. -200 degrees Celsius, and there is the obvious disadvantage that you are in ethanol won't evaporate away immediately upon removal from the bath. However, it might be fun to play around with. At the very least, you could probably make some interesting frozen alcoholic fruit with it.
Presumably, you're having difficulties finding liquid nitrogen. I recommend contacting someone in the local universities chemistry department. Tell them you are making liquid nitrogen ice, which is easier to explain then 'cryosearing a hamburger patty'. It is, generally, not heavily regulated, and with a little luck, I'm sure you can find some.