Hmmm, there are a number of variables there.
Generally, "fabric paint" is acrylic paint that's been mixed with clear textile medium to make it more supple on fabric once dry, and in general it's a little more waterproof than regular acrylic paint. Not sure how that would affect things on clay but fabric paint may be "thinner." Regular acrylic paints also come in various thicknesses with the cheapest ones in bottles being the thinnest and the more expensive "artists'" ones in tubes being thickest (also give the best coverage when "painting on" clay).
Dimensional or "3-D" paints are acrylic paints that have had a thickener added to them so they'll keep their shape while drying I believe, but not sure what kind of filler. I do know that one clayer suggested mixing 6 tsp. of baking soda to 3-1/2 or 4 tsp acrylic paint to create a dimensional paint that could be squeezed out of nozzles. (If it's just the 3D effect you want though you could also use thinned regular clay or thickened liquid clay to squeeze out for "piping" or like is done sometimes with mini foods, but I'm assuming you want to use the dimensional paint you already have.)
Then there's the issue of adhesion. Some baked clay surfaces are "toothier" than others so may work best (or not) ...I'm a little surprised they said it works on metal (I'd like to know just how well
it works on very smooth, non-porous surfaces like that, but polymer clay should be similar).
One clayer used a dimensional paint once, I think, to hold some feathers upright on baked clay. She said it worked well but she had used a glitter acrylic paint so not sure if that acted differently. She said that she could scratch it off with effort
but otherwise wasn't a problem (so don't know how well it would stay on with stress/abrasion).
To increase adhesion if necessary, there might be some things you could try:
...try Sculpey clay (very toothy, especially the plain white boxed stuff), but since it's not strong perhaps just use a layer of it over stronger clay
...prepare the clay well (wipe with alcohol or soap and water to remove any oils)
...prepare the surface by sanding very lightly (and removing dust well)
.....or apply a layer of liquid clay, acrylic finish like Varathane (gloss, satin, or just matte) or Future, or even acrylic paint or gesso, to the baked surface first (then probably dry or cure before applying paint)
...make the acrylic paint more adhesive by mixing with a bit of Varathane or Future, etc.
..."re-heat" the paint after it dries on the baked clay (be sure it's totally dry first since that could matter for some paints)
...coat the finished painted item with polyurethane (or dip it), or even coat with a clear epoxy resin or glue, to give a mechanical hold as well as an adhesive hold
It would probably be good to do some test scraps first using one or more of those techniques, then just see which works best.
There may be more ideas or factors on my Paints
page, under the first few sub-categories of the Acrylics