My thoughts are that you're having the right thoughts.
The only thing I can say with near-certainty is that the skip sounds like a good thing for your daughter, though it won't solve all her problems.
Our son, in kindergarten, is in a similar spot, though your daughter may be a little more academically advanced, hard to tell. The school is going to give him a three-week trial in a first-grade class starting next week, then hopefully will be skipping him to second. They plan to add enrichment at least for math and language arts, but I don't know yet exactly what that will look like. The principal told us that in our district, the rules prohibit more than a single skip within a school! Our son is on the small and young side, but fits in with just about everyone he meets; he also could easily fly under the radar, like your daughter. Right now I remain worried, but am hoping for the best and grateful for the skip and the attention that's being given to us.
Can you ask for MAP testing or something else to figure out her progress against the local math curriculum? One worry may be that she has holes in her knowledge. Documenting that she is at third grade level for math may help avoid getting second-grade concepts taught to her and boring her. Other ideas which occurred to me are to ask whether they can do curriculum compacting (where pre-tests out of a concept she already knows and goes on to the next block of instruction) and/or online stuff like EPGY
in the classroom. Sounds like you may be already thinking along those lines.
A skip to third might well be ideal for her. I'd personally get another skip while the getting's good, if she's on board with the idea and you have a strong feeling she would adapt socially. (ETA: Nice point by st. pauli girl below about handwriting too.)
If you haven't done a DYS application yet, are you planning to? The deadline is the first of the month to have it to them, and you need a letter of recommendation in addition to your own documentation.
