Does such filter exist?
It is not really a convolution, more a weighting. Your expression is $0$ homogeneous in $x$, as you divide $x$s values by $x$s.  Specifically:


it is not defined if $\sum_{-K,K} x[i+k] = 0$
if $x[k]$ is locally a constant, the output does not depend on it (because it will be equal to $\sum h[k]$)
if $h[k]$ is an averaging filter ($h[k] = \frac{1}{2K+1}$), the output will be $\frac{1}{2K+1}$


So globally I would not consider it as a  classical filter for $x$. But it could be seen as an adaptive weighting for $h$, based on a unit-normalized window $\chi$ with $\chi[n] = \frac{x[n]}{\sum_{|k|\le K} x[n]}$. As such, it is easily implemented in Matlab or Python. So far, I don't have a use for it. What was your purpose?