I definitely think this is supported by canon!
Since I’m on a 10 hour road trip and have nothing better to do, I went through screencaps of the episodes of book 3 and I noticed that the number of scenes where Katara does housework drastically decreases after Zuko joins the gaang.
In The Painted Lady, Katara is shown getting dinner for everyone while Sokka goes over his (extremely long lmao) schedule.
In The Runaway, when Sokka, Aang, and Toph come back from their money-making scheme, they return to Katara cooking. And leave all of the groceries/supplies they bought with her, so it’s natural to assume she’ll be the one dealing with them. Then, in a later scene, Katara is sitting next to the cooking pot and feeding Appa, while Aang eats in the background.
In Day of the Black Sun, while Sokka goes over the invasion plan, Katara gets him a drink. (Not exactly a chore, but it does show how she is responsible for taking care of everyone else’s well-being).
In The Western Air Temple, Katara is shown getting meals ready for Aang, Sokka, Toph, Haru, Teo, and the Duke.
In the beginning of The Firebending Masters, the night that Zuko joins, Zuko watches the group in front of the campfire, where Katara is serving everyone’s dinner.
And that’s it. We don’t see Katara doing anymore chores in the background after Zuko officially joins the group, except for when she makes watermelon juice for the group in Sozin’s Comet (although that seems like less of a necessary chore and more of just a special treat she made for fun).
I only went through the episodes in the final season, but there are many more scenes of Katara doing household work in the background in the other two seasons, usually all on her own. The only chore I observed the others doing this season, besides Sokka keeping everyone on schedule, was shopping - in just one episode.
Even though Zuko hasn’t been with the gaang for long, he’s the only other person we see serving the group.
Especially taking into account his behavior around Katara, it makes sense that he would be helping her out with the chores.
In The Southern Raiders, he’s carrying their bag for her, and later, he’s the one doing the packing. He tells Katara she should get some rest - literally the only time anyone in the gaang ever asks Katara to look after her own needs - and it’s shown later that they switched places and Katara looks more well-rested.
Given that part of Zuko’s arc involves working in customer service, I’m sure he’s very familiar with doing chores without being told.
Purely speculation on my part, but in this scene from Sozin’s Comet, the group already has their food and are eating while Katara was off in the attic. My theory? Zuko served dinner. Since she was looking for pots, Katara probably still does at least some of the cooking, but it seems plausible that Zuko might help out with prep, serving, and/or cleaning.
Unlike the first half of book 3, where Katara is often doing chores in the background, after Zuko joins, she has much more free time and seems more relaxed. She watches Aang and Zuko train (with Toph and on her own) and goes surfing at the beach.
And this just fits in with Zuko’s personality. You can’t convince me that Zuko - the boy who is so set on making a good impression on the gaang, especially Katara - wouldn’t help out whenever he could. It probably drives Katara nuts at first because she feels like he’s trying to take over “her territory,” but slowly she comes to appreciate his presence when she realizes that he just genuinely wants to make things easier for her.