Based on what you've said is on your pragmatic list and your wishlist, I'm going to add a few questions to your list. (If I've misunderstood which items below on which list, my apologies!):

-Does the district allow partial homeschooling? E.g., attending school half-days (regardless of k or higher placement) or on certain days of the week, attending only for certain subjects (say, math and specials (art, music, gym) only).
-Is kindergarten mandatory? (IOW, if there is no formal provision for partial homeschooling, could you reasonably make an informal arrangement, where she just has consistently bad attendance every Tuesday and Thursday.)

And in response to some of your existing questions:
-early entrance to first grade may or may not be a good fit--but it's quite difficult to accurately assess this, to be honest, at this early date. It's only February now, and, as you've no doubt noticed, a substantial amount of development can occur in six months. You do need to start conversations somewhat early with schools both public and private when it comes to grade skips or early entrance, but her EF is going to be a moving target.

We went through a similar conversation about early entrance to 1st with our DC1 (although for somewhat different reasons). In our initial conversation with one tiny private school, the director acknowledged, after administering their kindergarten/first grade screening/placement instrument, that DC was well above the expected cut scores for first grade (at around this time of year), but that they had concerns about maturity and safety (this was because DC's response to the stranger-danger question was, shall we say, in keeping with DC's high extroversion), and recommended kindergarten, with the usual line about how rigorous the curriculum was, and the ability of the teacher to differentiate. I observed the classroom subsequently, and learned definitively that it would be a poor fit for DC (not only academically--much longer story). The second tiny private school gave the same screening instrument again, a few months later, and similarly obtained comfortably first-grade ready scores, but also made the same comment about maturity, accompanied this time by concerns about high activity level. This time, we requested that additional data be collected at the end of the summer, both on academic skills, and informally, in DC's interaction with the teacher. At this third assessment (second with this school), the director pulled out some first grade end-of-course materials and early reading (think Amelia Bedelia, Frog and Toad-level), and established to her own satisfaction that DC wouldn't be out of place academically in second grade, which seemed to be what she needed to say that the tradeoff in "maturity" would be less than that in instructional appropriateness. It likely helped that additional time with DC allowed her to see that much of the chattiness and activity level were not purely developmental. She finally agreed with us that placing DC in kindergarten would not make DC less active, just active and inadequately challenged at the same time, which would be extremely likely to present as even more "immature behavior."

And the result of the early entrance to first was an excellent year, in which the very-experienced first grade teacher scaffolded all the EF needed to be successful (both behaviorally and organizationally). Because that's what first grade teachers do. Having older classmates to act as behavioral models also helped DC considerably, whereas (not unlike your DC's 3-year-old school year), having younger models would likely have resulted in downward social referencing instead--a point volunteered by the director, who was convinced within the first half year. It is likely that a child who has already demonstrated high levels of social perception, and the ability to adapt easily to the social norms, would perform up to the expectations of her first-grade peers.

Now obviously, your child, circumstances, and options are all different from ours, but just to say it can work, even with a supposedly "immature" child.
...pronounced like the long vowel and first letter of the alphabet...