Does a hurricane classification exists where estimated rainfall is taken into account?
At the current moment, none that are in operaational use. The problem with using expected measurements is that there may be multiple drastically different estimates, and then there is the problem of errors creating misclassification. There have been attempts, but none of them that are noteworthy or operationally used, that I know of.

This article may be relevant to your question. This article may also be relevant, but is obviously dated considering the events of Hurricane Harvey. Harvey also highlighted the problem of rainfall relative to runoff, which is significant in urban areas and concave/parabolic topography.