Twizzle currently aims for compatibility with
WCA notation, but I don't think FCN is a good skewb notation
outside of scrambling.
FCN by itself can only denote moves that keep one corner fixed, as its name ("fixed corner notation") suggests. This happens to work okay for scrambling (which is not to say some other set of four corners won't work better—I have not looked into this (because I don't care about skewb), but I assume at least some skewbers have, over the 10 whole years since FCN became the standard skewb scrambling notation), but for writing out algs, FCN is underpowered. Even the extremely commonplace sledgehammer cannot always be written in FCN!
Twizzle's current notation extends FCN by adding "UL", "UR", "D" and "F" for the four remaining corner turns. This preserves compatibility with FCN, but is now unintuitive. Granted, assuming you had no prior knowledge of any skewb notation, if I gave you a cube and told you that you had to label the eight corners with "U", "L", "R", "F", "B", "UL", "UR" and "D" in some order, then Twizzle's labelling will be the only one that makes sense; however, this would only be via a convoluted series of eliminations.
(Contrast the colloquial megaminx notation, where the top hemisphere's faces are "U", "L", "F", "R", "BR", "BL" (possibly with varying capitalisation on BL/BR), and the bottom hemisphere's faces are often "D-something". The faces that are easier to access don't get a "U" prefix (except the topmost face itself), while the faces that are harder to access get a "D" prefix. Note also that Twizzle's megaminx notation
doesn't follow this convention!)
The same labelling is also used for Twizzle's Redi Cube notation, but as the Redi Cube is a much more shallow-cut corner-turning cubic puzzle than skewb is, FCN doesn't make sense to use on it, and neither does Twizzle's current extended FCN.
Anyway tl;dr I think Rubikskewb notation is better for both skewb and Redi Cube outside of scrambling. (For scrambling, I don't have any qualms about FCN for skewb.)
I don't know what a good way of implementing multiple conflicting notations is. There are also conflicts between community-preferred notation and Twizzle/acn-supported notation for other puzzles, notably big cubes: bigblders prefer Singmaster/Singmaster-like notations, but that conflicts with SiGN, which is more generally useful.