I'm not sure what your idea is, so I don't know if it will work for the LED strip or not, but generally CCLs don't dim well. What tends to happen as the voltage gets reduced is that some fraction of the wire emits, and the other part doesn't, with the part that doesn't emit getting smaller as you dim it further. It looks like the only part of the CCL tube is lit, and you can see more or less light as you increase or decrease the voltage.
That said, I had a potentiometer laying around and I used it on my last build to dim a CCL. I needed to dim it just a little bit so that I didn't overpower some of the other case lighting.
So I wired the pot in (on the DC side of the inverter - i.e. between the PSU and inverter, not between the inverter and CCL). What I found was that if I just increased resistance with the pot a little, the CCL dimmed enough for the purpose I wanted. Then if I went further, it started doing the part-of-the-tube not lit thing, and then quickly went out.
I don't remember the resistance of the pot i used, some 10 or 20 k Ohms I think. And I guess if you don't mind how half a lit tube looks (e.g. can't see the light anyway), then that would count as dimming too!