The short answer to your question is yes. There is such a space, that of the locally smooth functions on $K$. Schwartz defined the space of distributions with support in $K$ not directly by duality but by defining the space of distributions, then the notion of restriction of a distribution to an open set and finally the support of a distribution as the complement of the union of the open sets on which it vanishes. In parallel he defined the space of distributions of compact support as the dual of $C^\infty$. It is then a theorem that the two concepts coincide.
The answer to your question is rather more subtle. This can be seen in the simplest case, i.e., where $K$ is the origin in the real line. The space of distributions with support there is infinite dimensional--it consists of the linear combinations of the Dirac distribution and its derivatives. This can obviously not be the dual of a space of functions defined on a single point. It is, however, the dual of the space of germs of smooth functions at the origin.