I'm in a similar situation OP. DP moved in with me, and my 2DC, he has none of his own.
We haven't merged finances.
We pay bills equally.
I pay a bit more for food.
I pay for anything else my DC need (clothes, sport equipment etc)
He WOTH, I WFH. He also works longer hours. So I often do end up doing a bit more housework, just because I'm here. But we both end up with similar amounts of leisure time, and we cook as equally as we can, with him working shifts. Whoever doesn't cook, cleans up. DP also helps with the school runs, shares the laundry, and has some set jobs that he does when it suits him, so that I don't even have to think about them.
We did have some thorough chats about this early on though. My exH was a lazy manchild and I wasn't risking that shit again. If DP didn't see us as a team, I don't know I'd have let him move in.
OP, I think the money side of things is probably fair at the moment, because you aren't married, nor have you merged finances. But the balance of chores is NOT. Your DP doesn't get a 1950's style housewife unless 1, he's financially providing for her and her DC, and 2, she's ok with that set up. He needs to step up, and split the responsibilities fairly.