Lola is clearly the favorite child (allowed to seek her own marriage). Thus her comments are based on the assumption Big Mom hears the Lola connection. As you addressed in the second paragraph, the "power" is only Homies specific. Thus in terms of the ultimate resolution (getting Big Mom to agree with letting Sanji go is the biggest thing that needs to be negotiated), Lola will be necessary which is also why she hasn't appeared yet. If Lola appears now, it literally allows for resolution of most, if not all aspects of the arc. To expand upon the vivre card discussion, the "life force" in the card is indirect. It is made from a nail clipping, so it is "part" of Big Mom but not part of her main "soul". The Homies can not defy Big Mom since she is their "creator"/"mother". Essentially the vivre card is the lowest level "soul". Next on the scale would be the "collectors" (forget if the black collectors were named). Since they are part of Big Mom's soul, they are able to have additional capabilities/supernatural events. The highest level would be Big Mom herself and the full "life force" manipulation. Her fruit was originally supposed to be the "Birthing Birthing" fruit, but was changed due to former editors leaking the information almost a year ago. As for the weather abilities, already posted that I do not see the sun and cloud Homies being the power (so not going to rehash it here). I personally see it as a second ability, but we will just have to wait and see (it probably won't be revealed this arc).