I think that even if she complains to her mom (I bet that Mom gives her attention when she does so - and she is lacking in maternal attention) that DH should continue to enforce the chores. If not, he's more afraid of teen attitude and his ex than he is the ire of his wife. There are a lot of times when a response is just not necessary. So if the mother gets all bent, just ignore it. It will drive her crazy. No CPS worker will take SD out of the home because she did a load of dishes.
My SD has slacked on her chores lately, too. I don't do them. If DH complains I say "You expect me to do x loads of whatever and take the trash out, but you have zero expectations for your daughter. If her chore is supposed to be to do a load of dishes and take the trash out, then she should do them or you should." He likes to claim that I forget that they are "only 17 and 22" and I remind him that they ARE 17 and 22. They are fully capable of the chores requested without reminders. I pick up after the 3 yr old, not those who are old enough to drive.
Further, I used to let SD pick to unload or load, knowing she hates loading. But now I unload and she gets to load. If she was quicker, she'd unload. I'm tired of getting the short stick.
If she has no life skills, she'll learn the hard way.
(There are many reasons a kid wouldn't live with their own mother. Sometimes it's a choice between the parents and sometimes it is that the courts have determined that the father is the more responsible or stable party. At the end of the day, this SM is dealing with a teen with another household to complain to and the usual teenage laziness that many people find frustrating.)