The Institute for Justice - https://ij.org/ - a "libertarian" ACLU
I consider their core focus to be on issues that are universal and non-partisan like Free Speech, Police Abuse, Civil Forfeiture, while also veering into more partisan issues like Educational Choice. I haven't found any truly non-partisan organizations like you are asking for, but feel that the IJ fills in some gaps in cases that ACLU used to or should be focusing on, and the areas where I disagree with it "balances" out the areas where I disagree with the ACLU.
The Institute for Justice - https://ij.org/ - a "libertarian" ACLU
I consider their core focus to be on issues that are universal and non-partisan like Free Speech, Police Abuse, Civil Forfeiture, while also veering into more partisan issues like Educational Choice. I haven't found any truly non-partisan organizations like you are asking for, but feel that the IJ fills in some gaps in cases that ACLU used to or should be focusing on, and the areas where I disagree with it "balances" out the areas where I disagree with the ACLU.