> Is there an impeachment process for a constitutional violations

There's an impeachment process. Creating a separate, faster one for constitutional violations is just inviting trouble.

The declaration of martial law was already unconstitutional in the part that forbids something that is explicitly mentioned in the constitution (a meeting of the Parliament to request that the president lift the martial law). Which is probably why the members of the Parliament only has minor obstacles towards having their meeting.

>I don't even know if there is a faster route than impeachment in the US system.

There's section 4 of the 25th amendment, but it is untested.