Tool to slice LED stripe lengthwise
If the LED strip is truely unobtainable in 10mm, then I'd be surprised.  Try checking other sources, like ebay and aliexpress etc.

If I was really stuck, then I would make a cutting block similar to those used to spiral-cut PET bottles into long strips.

You would want two razor blades mounted parallel and vertical, and a 12mm slot to guide the 12mm strip in straight.
-------------========= 1 mm waste strip
|   12mm     &lt;--------------  10mm wide LED strip from the middle
|   wide                        This line is the 10mm output strip
|   LED      &lt;--------------  Pull to the right by hand
-------------========= 1 mm waste strip
             .
             .
         Two razor blades in plan view, sharp edges point to the left.