Sounds like a great compromise: keep playing with sticks, but redirect her to something I want her to bite (instead of us!). Keeps our options open for the future while boosting her confidence and teaching her right from wrong.
If she still nips us at the start when using the rag/alternative, I guess we still stop play immediately (regardless of pet or Schutzhund)?
Don't play with sticks. Play with the rag or tug.
If you make the tug play exciting and rewarding, she shouldn't feel the need to nip at your clothing, she should be able to focus her energy on the tug. Tease her with it, make it move like a wounded prey animal, to pique her interest and focus. You might try getting a BIG tug toy so that she has a lot of surface area to aim for, and you can sort of block her with it if it looks like she's going for your clothing.
For a SchH-bound dog, most trainers advise not to train the "out" until later. But since she is primarily a pet, teaching her to "out" at this stage isn't a bad idea.
Do have a trainer who is experienced in SchH help you out with this, as there are many subtleties to tug play and teaching the "out", and small things can make a difference.