First, it sounds like you would not really have a fight with the school for SSA or full-grade skip, since the school has already said (and demonstrated) that they are open to either/both with documentation from formal testing. So I don't think it's so much a question of persuading the school as it is of determining what makes the most sense for your DD and your family.
I can't speak fully to the BTDT, since we've taken every skip offered, and then switched to homeschooling, but I will note that we advocated for early entrance to 1st grade a month before school started, and received it once the school had the (very reasonable) documentation that they wanted. (Not out of the blue. We'd been in discussions for four months before that.)
My bias generally, for a student who meets objective (e.g., IAS) criteria for grade skipping, is to give a fair amount of weight to what the student says she wants. But you know your child best, and how accurate her perceptions usually are.
