Ooh, what a fun little idea egg!
Let me see...Marluxia would probably have to rework his plan a bit, but we know that Kairi's princess/light powers didn't prevent her from 'forgetting' Sora via Sora's altered heart state, therefore Namine would just as easily be able to manipulate her as anyone else.
However, Kairi seems to have a knack, as it were, to kind of see peoples' 'true selves' despite their physical appearance. (Sora's heartless and to an extent, Xehanort-Riku-- albeit being a bit surprised at first, she very quickly figured it out). I'd imagine she would actually be pretty spot on at figuring out that Repliku wasn't real Riku. We know that the connection Sora and Kairi has is very deep and it took Namine a lot of chipping away before it finally got covered up. Sora met Repliku way before that point, so unless Namine hyper targeted Kairi first (which I'm not sure would have been possible)...she's going to quickly figure out something is up. That would throw a huge wrench into Sora's early 'motivation' to keep going deeper into the castle because it would be a bit obvious something is up.
Presumably this would then set into motion a few things. It would rightly establish that Kairi is a threat to Marluxia's end goal and if he hadn't already decided on it already, would extra motivate him to target her.
The only reason Marluxia likely never bothered with Donald and Goofy (and Jiminy I guess) is because they wouldn't really be much of a threat to stopping Sora from prolonged exposure to Namine (aka staying in the castle), but Kairi could potentially dismiss any/all of Namine's early lies because she was actually there. Though, I'm not sure the extent that Namine can construct fake memories in other people, but maaaybe she could plant the same memory in both of them like she did with Repliku?
Conclusion: unfortunately, unless Marluxia is very dense or Namine is greatly overpowered, i'd bet Sora and Kairi would be getting separated again at some point. Either Marluxia tries to kidnap her or whisks her away during whatever the hell kind of magic turned Donald and Goofy into cards. If they were together, I doubt Namine would have as much hold over Sora's memories/thoughts of her. That's the only way castle oblivion would be able to play out similarly.
Though I wonder if Marluxia would just consider trying to assassinate Kairi right off the bat. I mean, the whole reason he is even using Namine in the first place is because he knows Kairi = Sora's light. If she were to die at this point, presumably Sora falls into the darkness having lost his hope and would be easier to manipulate...
But that's dark and I don't wanna think about that.
My other major consideration would be if/when Kairi would get access to her Keyblade. That's still a bit of a weird point even in canon, but we know that theoretically she should be able to use it at any time so I suppose she could technically figure out how to pull it out earlier. If that happens, perhaps Marluxia would actually target Riku instead?
Rather than just have one puppet, why not try to have two keyblade puppets, right? And both Kairi and Sora would be motivated to try and help Riku so maybe instead of dangling 'new girl Namine' out in front of them he would actually try to sniff around the basement of the castle to find Riku and/or get a new angle that way.
Since part of Marluxia's plan was replacing Namine as the girl important to Sora I'm not sure if that would work with Kairi currently with Sora. The whole plan of controlling Sora would have to be altered to a degree
In regards to what weapon Kairi would have wielded in CoM I'd like to think she wouldn't get a Keyblade just yet. Rather like Riku she would get a weapon that could potentially become a Keyblade at a later point. I could see her with a rapier.
I've designed a story for it. Here's my stupid story. The story would consist into 3 parts: Sora, Riku, and Kairi. I feel the best way to get Kairi's story going is to separate her from the group. Sora being lulled into Castle Oblivion and Donald & Goofy follows. Someone needed to keep watch of Kairi to make sure nothing bad happens to her........................so Tinker Bell stays with her. (Now, when I first designed my KH1 story, I had intentions of Genie & Tinker Bell joining Sora's party, rather than secluding them to summons, since technically...they're the only summons in KH1 that aren't summoning gems. Technically speaking, they're in their party.
Anyway, Kairi wakes up, sees Sora, Donald, & Goofy gone...and she tries to follow them with Tink by her side, which leads her to Castle Oblivion. Clearly, if Kairi was there, she'd be detrimental to Marluxia's plan to overthrow the Organization. Also, while we're at this, we have to think about something here...Sora & Riku's journey in CO consisted of the big bad, their challenge.
Sora's challenge is to defeat Marluxia & protect Namine. Riku's challenge is to defeat Ansem & conquer the darkness eating away at him. So, should Kairi have a reason to be there, what do you think her challenge would be? Who would she have to fight against where Sora got Marluxia & Riku got Ansem?
If Namine was born via Sora and Kairi shouldn't she be able to mess with her memories too? Maybe Marluxia would want a PoH under his control?
It would be cool if Kairi would be the one to figure out Namine is her nobody. It gives Kairi something to do and allows them to flesh out Namine being her nobody. KH has themes of identity but I don't feel like Namine gives me insight into who Kairi is. Same with Roxas/Sora, Ven/Vanitas or the Xehanorts outside of maybe Young Xehanort.
About the keyblade- you could have her get it here?