I'm taking a page from
@Tartarus' book and am going to try and write down what I'd have done with her character. Though I fear mine is a bit more fanfictiony, though I'm going to try and keep things mostly in line with how the Dark Seeker Saga unfolded.
KH1: Mostly unchanged. Small changes Including that she's an optional party member on Destiny Island, giving the player a feeling for having party members. Sora would drop the Kingdom Key in HB before picking up the Keyblade of Heart and Kairi would pick it up as she runs away from Ansem, this should be a minor hint at her being a wielder and she'd return to HB with the Restoration Committee seeing her staying alone in Traverse Town makes no sense. She'd also have optional dialog stating how the castle is kind of familiar and how there's a slight unease being there as well as dialog with her an the other POH.
CoM: Unchanged other that I'd have Kairi and Sora appear sporadically in Riku's memory versions of the worlds hinting at them always being there with him wherever he goes.
Days: Midway through Kairi would go off searching for Riku, seeing he hasn't returned home. She would also have her awakening and gain her keyblade in this time period. She'd travel via gates she creates with the Keyblade, but wouldn't have mastered it too well ending up in random worlds. Maybe arriving on worlds as Roxas and Co. leave, having them constantly miss each other.
Mostly off screen stuff, we'd mostly hear about "The Princess" being on the on the move and stuff like that. With the Org discussing ,if she'd be a problem.
KH2: I think Kairi might have been to a few worlds Sora visits, but have left before he arrived. Like the HPO, would know her and let Sora know in his first visit to their world that Kairi's out there looking for him.
I like
@Tartarus' idea of Kairi being at the 1000 heartless battle in Radiant Garden, maybe as of that battle she'd tag along for the second world visits giving the player a feeling of progression, he was looking for two friends and has found one. Alternately in the climax of the battle of a 1000 heartless she could be whisked away eventually ending up a prisoner of the Organization.
I think she'd either be a party member in the world that never was
(along with Riku) for the whole thing or at least for parts of the battle against Xemnas.
BBS: Pretty much unchanged other than in the Secret Episode we'd get flashes to Radiant Garden showing us it's fall, how Cid took Leon, Yuffie and Aerith into his Gummiship and left and how Kairi was thrown into the Ocean between worlds and Aqua's magic saved her and took her to Sora and Riku. I like the idea of the two boys finding her after she washed ashore.
Coded: Part of me feels it'd be more logical for the Journal to take her form, with Data Kairi being incorruptible by Darkness. Data Riku would be working as her Guardian.
DDD: With how Kairi was developed as a Keyblade wielder now, it'd make no sense for her to sit out the Mark of Mastery. But with how DDD's story is set up it's hard to fit her in. I'd ideally want her to be a 3rd Player character, but the only way I could make that work is have her dive into Riku's dream, because she subconsciously thought he was after Riku again. I guess her Boss Nort would be Terranort?
KH3: Instead of training with Axel, I think we could have her and Axel look into alternative ways of getting Naminé and Roxas back. And they could be searching Castle Oblivion for the Chamber of Wakening, as a backup plan in case Aqua couldn't be found. They'd also be tasked with helping each other train. Maybe even looking for traces of Vexen's research.
