We did both private and public with our dds. We found that the public school here had more peers and was more flexible (the private was not a gifted school, just a parochial school). However, a LOT depends on the teachers and principal-imo, the principal is the most important because a teacher is often willing, but afraid. Once the principal okayed it (it being going to another grade, alternate spelling lists, expanded projects, etc..) the teacher almost always agreed. This isn't to say we were meeting with the principal every week, but we sat down with him and the teacher(s) at the beginning of the year (and provided above-level achievement testing to support our case)and mapped out accommodations.
I also think that home enrichment can accomplish a lot. If you could put the money towards some awesome summer programs, a few classes, etc... you could really fire up your dd's love of learning. Also, how long will the parochial school stay ahead? In our case, they were ahead until about 3rd grade, when more grouping and differentiation started (separate from the gifted program) at the public schools.
Obviously, though, this is all completely dependent on your specific schools. I'd definitely schedule a meeting with both- talk about the grade acceleration and curriculum at your current school, and what accommodations could be made at your public school.
Just fyi... we found that a grade skip was much easier to accomplish at the parochial school (and they were incredibly supportive!). So, another option is to grade skip now at the private school and then switch in a year or two to public at the higher grade. Many public schools seem to be anti-switch, but they are fine if you come in with it already done.
