I want to try and outline the cases for Carrot and Yamato the way I see it.
Lookout is fine for a role for Carrot. Franky was the helmsman before Jinbei joined, so I don't see how other characters doing that duty is a problem. As far as Yamato, she was introduced during the raid and it's been fighting and plot exposition even since. She could very easily have a role defined later, but she doesn't have one now.
As far as dreams, they both seem to have the exact same one, so it's not really a point for or against either of them.
From a relationship with the Strawhats perspective, Carrot clearly has a good relationship with Chopper, a lot of the other strawhats know her and Luffy at least knows her name. I don't think she really needs any more to qualify, but it's certainly not a strong point like with Jinbei. Yamato is pretty much unknown, but again, she was introduced during the raid, and it looks like she is going to bond with Luffy very soon once he comes back to take over for her against Kaido. It's not an area where she has enough development to be a Strawhat yet, but it can still be developed since she was recently introduced and has a prime opportunity coming up.
At this point Carrot has a huge lead just because her points are established while Yamato's are up in the air, but it is subject to reconsideration as we see more of Yamato.
Now the plot/formula perspective is where Yamato takes the lead. Carrot fits all the other parts better, but she falls very short here.
No arc centered around her, no personal villain or major struggle resolved by Luffy, no major backstory, no "Luffy is the man who will become the pirate king" or some equivalent moment.
The pirate king moment can still happen and really isn't a problem due to how easy it would be to put in, but I think the others are long past the point of being reasonable to expect. Zou should have been her big arc, or WCI, but they weren't, and she has had very little spotlight in Wano. Her only personal villains are Peros and Jack, and they seem to be being taken down by Inu and Neko. Finally, it seems her backstory was the short one with Pedro, which is far too short for a Strawhat.
Yamato doesn't have all of them checked off either, but she is in a better position to do them. She had her pirate king (equivalent) moment just recently when she confronted Kaido. She has her big story arc going on right now. Her personal villain is obviously Kaido, who Luffy is positioned to defeat. The backstory isn't checked off, but could easily still happen.
As far as miscellaneous things, they both have points in their favor.
Carrot has a unique Su long form that is completely different from the other minks including the dukes.
Carrot has traveled with the Strawhats for two storyarcs, and is often grouped with them, though not always. She is never grouped with the other Minks.
Yamato has already declared her intention to join the crew twice.
Yamato has a strong connection to Ace.
If we pretty much put the miscellaneous things as equivalent, which is debatable, and we assume that whoever joins will complete the strawhat checklist, then it basically comes down to which of these scenarios is more likely.
1. Carrot gets a pirate king moment, Carrot gets a character centric arc, Carrot gets a bigger flashback, either Luffy defeats Perospero or Jack or Carrot is revealed to have another unknown personal villain.
2. Yamato is revealed later to have a specific role, Yamato bonds with Luffy significantly more than she already has, Luffy learns Yamato's name, Yamato gets a tragic backstory flashback, and Luffy defeats Kaido.
Personally I think scenario 2 is much more likely. I'm a fan of both characters, and ideally I would like both to join, but I'm trying to make an objective analysis of the two.
Side note- as far as Carrot's character centric arc, it doesn't have to be the same arc she joins in. Nami and Robin had their arcs after they joined, and Jinbei didn't join until several arcs after his.
Another side note- As I wrote this all out I realized that Law fits the Strawhat checklist significantly better than either of them. He has other things going against him but make of that what you will.