Clearly, she needs more than she is getting in first grade currently. Her teacher already recognizes that she is very bright. This is very important, and I think you should take advantage of this before your dd "checks out". I agree that you need testing, but I think you could make a case for in-class differentiation at the very least right now. I would bet that if you e-mailed her teacher back now while she is still blown away by your dd, telling her what you just told us, including how upset your daughter is and that she is crying at home over the lack of learning new things, that the teacher might be willing to at least do some assessment to find her approximate instructional level. Does the school use the NWEA MAP? If so, the regular MAP test (not just the Primary Grades test) might give valuable data that would allow the school to see what she is ready to learn next. If you want to walk in with some ammunition without asking anything of the school, and without spending a ton of money, you might have your dd take the DORA and the DOMA at home and bring in the results.
I hope this helps.