How about putting it all together and treating the stains and washing the whole quilt after it is completed?
I know what you mean about cats. We have Abby who we love, but she barfs a lot. Can't keep the carpet clean. We've tried putting her on special food, elevating her food bowl, etc. but she barfs at least twice a week and it isn't because of hairballs. We just put up with it and try to clean up as best we can. It does get a little frustrating when you have your carpets professionally cleaned and a few days later, there are barf stains.