Eh, just ignore. Raise your DD the way you would with SD and MIL, and forget about her silly competition.

As long as they aren't expecting you to pay for anything or run SD around, why does it matter?

At the end of the day, both girls will get to take swim lessons and both girls are happy.

Really it seems to me that you should concentrate on your dd6 and let your OH get on with it with sd6 himself, that includes purchasing her necessities including clothes. Having read your other post it would bother me that a fuss was made about poor widdle sd's lessons when you mentioned wanting your dd to learn to swim, the fact your OH was adament that both girls were to be treated completely equal then he colluded with MIL to set up swimming lessons for sd6 thereby excluding dd6. It was also incredibly bad form to allow sd to act like a brat by rubbing her lessons in dd6 face, when you had postponed dd's lessons to accommodate OH wishes and in consideration for sd6 feelings! THAT is the issue, your OH. Your sd is being taught to be unnaturally competitive, I get that kids do this to one another,but its the adults responsibility to shut thay down and teach the child the right path. Again sd's attitude is demonstrated when she is disappointed to learn that dd6 in fact is going swimming. What has OH said about that? Nothing I am betting. She sounds a brat and your OH is lazy.

Sorry to say and quite childish on his part if he is particpating in a one upmanship with MIL against you, and the fact he appears to say one thing and do another tells you everything you need to know, whild I am sure you want a family unit where everything is relatively fair and where youcan trust what OH saying, but imo you cant. OH has demonstrated that, he said nothing and did nothing when these issues arose except to enable his mother and sd6. This is a silly problem if you think about it, but this small issue just demonstrates the ultimate problem, that OH does not have your back and is willing to throw you and dd under the bus to placate MIL and sd!

So stop enabling OH to continue living in lala land, where you are held to a higher level of rules, expectations which can change depending upon OH mood and his family and he can down right dismiss his own stepkid= dd6 and his partner and where he can ignore the very same rules HE insisted upon! Stop buying things for sd, that his responsibility from HIS own pocket, as I am betting he doesnt have the same consideration for dd (hell he doesnt for HIS own daughter!). Stop communicating with MIL, screw her and her rudeness. If you stop engaging in HER competitive behaviour then she can only bleat and whine to OH and I would be ready to stop any and all that type of behaviour. So next time OH insists you have to include sd in something you want for dd, tell him "you are her dad, so that is up to you. Given the last utter debacle with the fake equality for the girls, I am abdicating from doing that. Its on you as her dad to do x if its that important to you"then ignore ignore. Expect manipulative tactics from OH and MIL when its comes to this, they will pull at your guilt in order to get you to re-engage but ignore them, do not allow these people to trick you from doing these things for dd6, as all they are doing is cheating YOUR little girl out of the things she should have with you. She only has one mother and father, perhaps your OH needs to realise the same for his child, that you do not snatch from one family to feed the other and that sd is very lucky she has not only her parents but a good SM too, so in affect she has more than dd6 as MIL wades in when necessary also!