One suggestion I like is to totally change the test format every year. This would basically invalidate test prep and put everyone on the same footing.
The test changes every year? No problem, just prep for the meta test like how people prep for the FAANG interviews. Multiple formats? No problem, just spend more time preparing for each format. Changing topics? No problem, just cover as many topics as necessary.
I think the key is to see that the poor, by definition, have much less time and money to tackle the problem. Therefore, to have a level playing field we should design the test such that prep may help, but there is a ceiling to it, and I think the SAT is somewhat decent in this respect.
The optimal test strategy can be covered in one afternoon. The format is well known and cheap study guides/sample tests are available pretty much everywhere. If the test changes every year, imagine how much more difficult it would be for a poor student to prep for it.
Edit: typo
As for standardized test scores and ethnicity/culture, that is a sensitive and contentious topic that I prefer not get into.