I don't know your state laws, but in many places if the schools go to the trouble to identify students as gifted and have a program for them, they may be grouped under the special education umbrella for policy purposes. Sometimes that means that there are specific rules that the schools need to follow. You may be able to use those rules to request that the work that you send be a replacement for class work rather than additional.
I also found it a good idea to dig through state and school district policies, guidebooks, and handbooks. Much as they usually have some guidelines written down for when they retain students in the same grade, they also may have other written guidelines for that you may be able to use to your advantage. I've found it useful to use the school's or district's own standards, as they are written and show how they apply to my child to get what we need.
When we did homeschool days with my DD, we didn't tell the school. Given how miserable she was at school, I considered it a sick day/mental health day. I work full time, so I took those days as vacation - she knew from the beginning homeschooling was not a long term solution, but she needed something to keep herself together long enough to get through the end of the school year. Concurrently with these random homeschool days, we were also working on the grade skip, so there was a short-term solution and a long(er)-term 'solution' working together.