What makes the extract work is the alcohol that evaporates, leaving the color behind. You need something that will evaporate, and although I’ve heard of non-alcoholic extracts, I can’t find a lot of info about them. You could use the spray colors from PME that are newly available in this country (PME is an English sugarcraft company) but I’ve found them to be expensive ($10/can and you can only get one wedding cake out of a can) and I don’t know what they use for a propellant, it might be alcohol too.
If you are painting on details (vines, leaves, flowers maybe) you could get away with using airbrush color (it will eventually dry, but not too fast) or diluting paste or gel color and applying it sparingly. The gel or paste “paint” won’t dry, but so long as you don’t touch or retouch the area you could get by with that.
Painting on the entire cake, though, is a different matter and you’d probably need to invest in an airbrush. I know that CalJava (aka Sweet Inspirations) sells pearlized airbrush colors now if you are looking to get that satin-y pearlized shimmer on the whole cake.