There are no gifted programs in our school that she is currently in.
Are there gifted programs or guidelines for identification elsewhere in your school district? If so, I'd consider making a written request for her to be evaluated by the school district, which will cost you nothing.
If there isn't any method for identifying gifted students within your school district, you could talk to your pediatrician, let him/her know how your dd is frustrated by lack of challenge in school, and see if your ped would refer her for a neuropsych eval, then check to see if that's covered by your insurance. Getting an IQ test just to get an IQ test usually isn't covered, but if you have a referral from your dr it might be covered.
Do you have any options for private school in your area, and if so - is that something you can consider? We have put our EG and HG kiddos in private school (two different schools) and found it's made a world of difference in how much they enjoy school. Each school advertises that their curriculum is about one year advanced over public school and we've found that for the most part, it's very true. Neither school is specifically chartered as a school for gifted students, but ds' school both have academic standards that have to be met for kids to enter, and in ds' school the standards are high enough that most of the kids there are, I'm guessing, above 90th percentile in IQ. It's not the same as being able to put him in a peer group where all the students are above 99th percentile, but it has made a huge difference in how much he enjoys school! DD's school is more of a mix of abilities, but having a structured classroom where behaviors that are often present in her public school aren't tolerated and where parents are serious enough about school to pay tuition has made a big difference for her, combined with having more challenging work. She's still finishing work ahead of everyone else in class and subject differentiation isn't as good as it could be, but it is better than where she was - and for her, too, having that incremental "better" made a huge difference in how she feels about school. If there are private schools you are interested in but think you'd never be able to afford, ask about financial aid or scholarships - there might be a way to make it work.
Last thought - look forward to middle school - what is it like where you're at? Middle school here is somewhat better for kids who need challenge in school because it's where groups of students are tracked by ability and higher-achieving students are finally in class with only higher achieving students and class moves at faster pace, plus math acceleration happens for many of the high ability kids. It is also easier where we are to advocate for >1 grade level acceleration (by subject) in middle school based on proof of achievement - so we let our ds after-school in math and science (his areas of strength) as he wanted to during elementary, in a way that we could show proof of his ability and achievement when he entered middle school, and we were able to get subject acceleration beyond what is typical that way.