In taking the SSAT or the other private school placement test (can't think of the acronym), she is competing against the brightest kids in the world who are trying to gain admission to the most selective schools in the US, not just your local private school. Because the test is essentially curved like the SAT, her percentiles are going to be much lower than what she would receive on a standardized national test like the Iowas or a state public school standardized test. I know plenty of kids who score in the 90s on the Iowas, and struggle to break the 50s on the SSAT. The nature of the test is to differentiate those kids who score in the top 10 percent of the regular tests.
What materials are being used by the tutor to coach her? Often, the practice tests that can be purchased commercially are easier than the actual test questions. If she's nailing the practice materials and not the actual test, then I think the issue is your tutor and test prep materials, not test anxiety. Really, most kids that age don't have test anxiety. This generation is so tested in schools, that the process is routine.
If you don't know the score she needs for admission to the school in question, then I highly suggest you meet with the admissions person at the private school to find out the range of test scores normally accepted. You may be surprised that the scores will be lower than you think (unless of course, you're trying to get her into Exeter, Andover, etc).
I normally think the college confidential website is overwhelming but you can get some very helpful advice on its board for prep school admissions.
Good luck!