How to repair damaged outer insulation on non-metallic cable and BX?
As others have pointed out, you aren't allowed to modify a box as you propose in option 3, but there are open splice boxes for this purpose.  These are designed for enclosing illegal bare splices and made with slots instead of holes, so you could slide the damaged NM cable in without cutting it.  Here's an example of one: https://www.garvinindustries.com/54171-opr

I'm unaware of a version for BX.