I have a very similar problem as yours. Went to the Melbourne Flinders Lane boutique and my SA said she doubts chanel can fix it. Its 'wear and tear' but she said I could try my local cobbler coz they would do the same but will take 4-6 weeks so it's better I take it to my own one. TBH I think chanel should fix it, it's just pulling out the bottom sole and replacing with a new one. I don't mind paying for it, it's better than having to pay $750 for a brand new one again.
Thanks.
How's your espadrilles now? Did you take it to your local cobbler? I am kinda hesitant to do it in case they ruin it more