This time I will try to explain what role the cake will have and how this arc will end.
First of all, I think the “Carmel inhabits Linlin’s body” theory has not yet been debunked. I even think it is the case and here is why:
Devil Fruits are connected to the soul, this is evident due to them reincarnating which is a thing only a soul can do and fragments of Big Mom’s soul being able to use her powers.
Technically the soul DF powers should be able to be transferred if the user’s soul inhabits another body.
Now, how is the possession theory related to the cake? I think the cake will indeed knock out Big Mom and then Carmel will take over her body completely as before she couldn’t do it due to Linlin’s own will power.
As some speculated Whole Cake Island heavily mirrors Thriller Bark and just like Linlin mirrors Oars Carmel being inside Linlin would mirror Moriah being inside of Oars, thus Carmel would be Moriah.
What happens then?
Carmel will explain her dream and that she used Linlin all along to create the slave empire of all races. Then everyone will work together to stop Carmel and free Linlin.
*Theory by Flower