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I'd really like to know if the I-130 application for the dependant child can be done in the US because she is a dependant, or if I have to do it now as a completely separate application?
Does the child have a legal way to enter the US (taking into account the eventual desire to immigrate, which rules out using any tourist visa or VWP)? If so, the I-130 can wait until after entry, and can be filed with adjustment of status. If not, you may want file the I-130 right away, so the child can get an immigrant visa. That would also avoid the adjustment of status paperwork and hassle.
When the spouse's I-130 reaches the consulate, the K-3 is not available to the spouse anymore, so the K-4 becomes unavailable to the child. If this happens, it can leave the child with no way of entering the US, other than starting from scratch and filing an I-130 for the child to get an immigrant visa.
The child will eventually need the I-130, so you might as well file it ASAP.
The child can't "tailgate" and get an immigrant visa based on the parent's I-130, the way the parent's I-129F can be used to get the child a K-4 visa.