This is an interesting topic
Ax
For Ax, I think we could get a huge hint from mm4 and book 52. I would think one half would be an insufferable, egotistical snob that probably wouldn't even want to associate with the humans and do things his own way. Except for the food, maybe.
I think the other half would be a scared, shell-shocked kid (from, you know, watching his entire dome ship get blown to bits, his brother dying, and being stuck alone on the planet) completely dependent on Jake and the other humans for direction. However, I think this half would care deeply about the humans and be extremely dedicated to them because they take care of him. While I would guess that he would perhaps feel hostile for the Andalites abandoning him, I don't think he'd want to be human. He never expressed any interest in being anything other than Andalite, except once when he contemplated running off with Estrid, but that was more about escaping war and being with her than about being any particular race.
Jake
Jake I think would be half strong, ruthless, fearless leader. And he would do ANYTHING to win.
The other half would be extremely conflicted and guilt ridden, incredibly paranoid about the lives of others. I can't even seeing him being leader, he'd probably refuse to make any decisions.
Marco
For Marco, I think one half would be his joking, happy go lucky personality. He'd probably be done with the war and want to just run around chasing girls and feeding Ax weird things, morph a surfer dude or something like that.
The other half would be kind of like the first Jake, but more a cold and calculated and brilliant strategist. I would think he'd be almost totally devoid of emotion, since most of his emotion is either comprised of or covered up by the first personality.
Tobias
Tobias, I'm not sure, that one is rough. I think one half would definitely be human and go with Rachel, learn to deal with his human past and be happy with her.
Honestly, I think the other half would want to still escape human life, but go be Andalite with Ax. I think the one book with Jake where he morphs Ax and stays Ax is a huge tip off for that. Also, he had so much emotion: mostly excitement and happiness when Ax suggested he morph Ax, and though for some reason he refused to let Ax know, he desperately wanted to know everything about Andalite culture. I don't think he'd want to stay hawk, but then again, I don't know why he didn't go Andalite in book 54, probably mourning Rachel.
Cassie
Cassie is another rough one. I think half of her would just go quietly along with the group, doing whatever they said, the guilt eating her alive.
The other half, honestly, I think would have left the animorphs and joined the Yeerk peace movement and tried to fix the war on that angle.
Melissa Chapman
(Based on book 2, 31, and MM3)
I've actually thought a lot about Melissa, because I'm currently in the planning stages of a fanfic series featuring her as the 7th animorph. Its rough researching and trying to portray her accurately, because there isn't a lot to go off of in the cannon. A lot of this is probably speculation and just taking things out of context (I have an entire degree in taking things out of context, woot English BA, lol.)
I think this split would actually be kind of sad. In the first book she is extremely depressed and down and out, and even then Rachel says there's still strength in her. But despite that she was like a total stranger to Rachel. I think if you took that strength out of her she'd be depressed to the point of suicide. Even though Rachel says she has strength, I think there would be a part of her that is mentally fragile and easily traumatized. Even though her parents are still around, she mentally fared much worse than Marco when he though his mother was dead.
I think the other half would have the potential to be a warrior personality, though not like Rachel's. There's the strength aspect that Rachel mentioned. Also, despite the fact that her parents essentially began neglecting her to the point she wholeheartedly believes they don't love her, and she has no idea why, she still is extremely worried about them and, in her own way, fights for them. I mean, I don't care if I had the coolest, best, most loving parents in the world...if a freakin gorilla tied me up and kidnapped my dad after trashing my house...I ain't looking for them, I'm letting the police sort it out. Yet despite the fact her parents treat her like crap and she's clearly scared out of her mind (since she was calling for mommy and daddy, and I don't blame her) she's running around town looking for her daddy. It takes balls to run around town at night, let alone after you just got tied up, your house got trashed, and your dad got kidnapped by someone when you probably weigh 98-115 pounds soaking wet. (From a former description of Rachel's in book 2.) I don't get the sense that she would like fighting just to fight like Rachel (because she is genuinely frightened and didn't tell off the gorilla or anything like Rachel would.) But I think that she loves her parents so much that if she had the chance to fight, she'd give it everything she had to try and free them, and she'd be much happier doing so like she seemed in MM3. (Though she got like, two sentences so its hard to determine what her mood was.)
Though, I definitely get the sense that she's a bit of an airhead too, from the description in MM3, where she is portrayed as never getting Marco's humor. (I mean, how hard can it be to get his humor? Its not particularly sophisticated.) I don't know where that would fit in, I can't see that fitting into either extreme of a personality, if she's depressed she's too depressed to be air-headed (like in book 2) if she's focused and fighting, she's too focused to be air headed...actually.. .ok...running around the street at night can be kind of air headed, but I think that would be more of a bravery/focus thing. I think as her mentally healthy self, she has the potential to be a real ditz. (Though that is probably just my head canon running wild, lol, well, its a fanfic, I can take some artistic license. I mean, if Rachel won't lament about how morphing makes plastic surgery impossible, someone has to!)