The way I see it you have two independent issues:
1. Should DD8 be tested?
2. How do you deal with the fall out of the results given the different personality characteristics of the children?
The answer to question 1 depends on whether there is a reason to test and whether anything will change for DD if the tests show she is gifted. From your comments you have a reason to test based on the issues she is having, bored at school, not doing the work, not having an easy time with the teachers. If she is having difficulties that are indicators for testing then she needs and deserves testing because it may well change her school experience for the better and avoid underachievement and acting out.
Once she is tested you have to manage the other potential problems. The concern over jealousy or competitiveness shouldn't dictate whether you have access to information necessary to guide your daughter's education.
It may be that the problems you anticipate don't arise. With a clearer picture of your daughter's needs, gifted or not, you can make her life at school happier and that may make her feel better.
