A 'flow-over' process of paint removal would be the best for this job. This would only be available from a shop specializing in this technique. Almost any paint remover woud have taken the paint off the cane seats if it had been sprayed on but it sounds like it was brushed on and maybe more than once. You could try TSP (trisodium phosphate) which is a brush cleaner. It's a mild form of lye.