Looking at BMX cables online last night I realised why it is that you had such trouble cutting through that cable outer: it must have been that compressionless cable housing (called linear cable in BMXland) instead of the usual coiled stuff. Compressionless outer is used on BMXs in order to stop the brakes from being applied when the rider spins the bars and stresses the cable. Compressionless outer is a bit like standard gear outer, in that it has wires running the length of the housing, but it's thicker and therefore very very difficult to cut. The recommendation is to use a dremel as suggested by 02GF74 above.
I'm not sure why your bike is set up with this type of outer housing as it's only necessary on bikes where the cables are prone to extreme stretching like BMXs. Anyway, I'm glad you got it sorted and apologies for doubting you.