My dogs have taken Clavamox dozens of times, and I've never been told to refrigerate it. The pills are kept in their light resistant packaging, but I've never been told to keep it in the fridge.
Clavamox can actually be stored outside of the refrigerated and doesn't lose potency or become harmful if not refrigerated for 2 weeks or more at moderate temperatures. If the temperature in your home is above 80 degrees it will lose potency quicker, but will not cause any harm.