I honestly don't think there is a way to construct a set of only caret, but if there were, I'd say [^-^] was it, since the set "everything but - and ^" can be constructed with [^^-] instead.
[^-^] gets extra bonus points for being humorous and logical at the same time. :-) Should someone feel deeply for this, my vote says it is the least bad backwards incompatibility to introduce to support it. :)