I see the biggest pros and cons of private testing in your situation this way:
+ You'll potentially get documentary evidence in testing you can directly use in advocacy, and no matter what will get some information that will help guide your advocacy efforts.
+ You'll potentially get information that will help to identify learning disabilities, where the earlier the identification occurs, the better.
+ You can pick a reliable tester, and decide which test will be used.
+ You don't have to wait.
+ You needn't share the results with the school if you don't want to.
- It will cost money, which might be a near-complete waste if the testing from the school in January is of high quality (assuming that IQ testing will be part of that).
- The school may not be as accepting of the results as if their own tester had been used (though you can increase your chances by ensuring that tests are used which the school finds acceptable).
- You'll have to be careful not to use the same type of test the school will want to use.
Pros and cons of testing through the school:
+ It will be free.
+ You'll potentially get evidence to help in advocacy, though the likelihood may be reduced a bit due to the chance of subpar test(s) and tester(s).
+ You'll potentially identify LDs.
+ Easier (possibly instant) buy-in by the school if the testing reveals giftedness or other needs.
- Increased resistance from the school if the testing doesn't reveal those things, and of course the school will be privy to the results no matter what.
- The tester is likely to be more familiar with testing for disabilities than for identification of giftedness, and may lack patience or the knowledge of gifted quirks that can help ensure a better result.
- You'll have to wait, entailing an extra delay of at least a semester in getting access to services, maybe longer.
- The school may use one or more suboptimal tests, e.g. the CogAT or another screening test which will give less reliable results than the WISC-IV, SB-V, et al.
