That comes out to about 6 mm, i believe. If this is correct, and you can find a 6 mm router bit, route a recess into the face of your neck a bit over 6 mm and glue it in with epoxy (other glues might work as well). You want it to be a good fit, but not so tight that you have trouble inserting it into the recess. When I did one, I made a slightly oversized wood strip to fill up the gap between the CF strip and face of the neck, which gave me something to clamp on. Then I used a handplane to plane that down flush with the neck. I haven't tried to plane CF as someone suggested in the thread you responded to, but I hear that it quickly dulls edge tools and creates nasty dust that you don't want to breathe.
If you can't find the right size router bit, you could cut it with a table saw (either dado blade or multiple passes with a regular blade). That's fine, except that you end up with an overcut at each end that you need to fill with a wood insert.
I'm not sure what the minimum dimension of the CF is needed to make this worthwhile doing. I used a .125"x.325" strip oriented 90 degrees to the face of the neck, which I believe is typical for what other people use for ukes. Yours isn't quite as thick in that direction, but it's double width. I would guess that if you try to bend it under hand pressure and can't perceive any movement, then it's likely stiff enough to do some good.