|Most likely it's polyester in general that she's reacting to (fleece is 100% polyester) so that cuts out pretty much all the usual stay-dry fabrics like suedecloth or athletic wicking fabric (I think this is what ME uses for their stay-dry fabric).
The only other stay-dry material that I've ever heard of that is not made from polyester is silk. I've heard of some using raw silk liners to help keep their babies dry. I don't know how well it works (maybe fairly well but not quite as good as fleece???) but it does make sense because silk dries very quickly. I don't know where you'd find these but it's something to go on at least...
You could also consider no stay-dry liner to see if she does ok with that. Not all children need stay-dry barriers, even at night.
Consider also perhaps the possibility that maybe it's a build-up in the fleece causing a reaction as opposed to the fleece itself. It's a bit of a stretch I know but fleece does tend to be more difficult to keep clean/build-up free so it's not impossible even if the diapers are ok. One indication could be perhaps an ammonia odor in morning diapers, even if it's light or tolerable enough that you haven't considered it an issue before.