Adding to Grinity, if school is still going on, have her do a day visiting at the private school. We recently had a friend's child (albeit, a 5 year old) whose behavior turned around completely when we moved from a bad preschool to a great kindergarten (at a private school, also struggling for enrollment in the recession) in the middle of the year.
I've been really shocked with how bad our state is with any kid who isn't average-- in my home state growing up, public school sucked with gifted but they were fantastic with other special education. The school we're sending DC to next year has a lot of both precisely because the public elementary has failed them. Depending on what state you live in and what you find, they may pay for outside testing, schooling, and/or therapy if they cannot accommodate you. But only in some states.
In your situation, based on what you've said, I wouldn't give up your job to homeschool. I would start exploring additional ways to save and make money so you can better afford additional help. And do those tests. I bet you don't have to commit to enrollment at the other school until August or September, despite what they may say. There will still be spots available, so if school is out for the summer you can wait to observe at the beginning of the year next year, and you can for test results.
Good luck!
