The impeach process is only the two step act of removal. House find him guilty of charges. Senate decides on punishment - removal if necessary
Any charges after that are seperate charges. WJC was found guilty of perjury - had license removed etc.
The process of original jurisdiction thru the federal court is a completely different check/balance. That has never been tested/used by the system iirc. But it is listed under III
The Supreme Court does exercise original jurisdiction in disputes between states but if anything, in recent years they have refused to exercise original jurisdiction more than they have granted it.
It might be worth a try though, nothing ventured, nothing gained.