Krusty's line, "I knew my kind wasn't welcome here", is likely a reference to his background, as many country clubs historically didn't allow Jewish members.
On the audio DVD commentary, it's mentioned Marge was allowed to say that she had a Chanel dress (both the discounted suit and the second one she bought when the first one was ruined), but they couldn't show the actual name (which is why Marge's fingers covered the label and a tree covered the Chanel storefront).
The country club women's clothes were changed in every scene, something Susie Dietter thought was hard to do because the animators had to come up with new designs.
This was the first time a female writer and director were credited in the same episode.
Marge's line "I wouldn't want to join any club that would have this me as a member" is paraphrasing a quote by Groucho Marx.