I think the question is do you think he would benefit from the gifted programming at your school specifically? If so, put him in to be tested. If the program isn't right for your son, that's okay too. You can also test, but then opt out if it's a poor fit. Regarding your daughter, think about it this way. Being gifted can be affected by the 2nd e - she should know that that doesn't mean she isn't the same person she was before she took the test. It's just a snapshot of where she is at the moment, and what her needs are. If the snapshot is saying she has attention problems, wonderful, now you can help her there. It's important for her to understand she is not a test score (which I am sure you are already helping her with!). But keeping him out of a program that would benefit him because it might result in self esteem problems later is borrowing trouble…
So ask yourself, do the benefits outweigh the risk?
After my experience with my daughter, I don't know if the school-based gifted programs have any benefits at all. One day gifted, the other not. Really?
We do a lot of outside enrichment, with summer programs at Northwestern University center for talent development, math programs (foreign approach), music etc. Shouldn't that be enough?