I tell my kids that above-level testing (SCAT, EXPLORE, SAT/ACT) is an "opportunity" for them to challenge themselves, not a forced requirement. You're the good guy, seeking & providing opportunities. I also see it as an interesting way to spend one Saturday morning, when they otherwise would have been watching cartoons, or doing nothing (fighting with their broters). Not a big deal, not much investment of time, money, etc. My H has come around, but originally didn't see why they'd want to do it. Same for the summer institutes--he thought, wouldn't kids prefer normal camp where they swim, play, campfire? We've done that, too, and it's fine, but the problem with it is that my kids are placed with agemates who can be annoying to them (not the same interests, not much in common). The real connection with peers is at the institutes, and my kids are hungry for that interaction with professors and peers & level of learning. You're just looking out for best interests of your daughter, trying to help meet her needs. The only way I have been able to convince H of this is by his witnessing the results. At first I just requested he let DS try it and I'd never ask it again. Now he doesn't have objections.
You may never convince your ex of this mentality, but don't get discouraged about seeking appropriate opportunities for your daughter. Maybe your daughter will be able to speak up for herself with him as she gets older.
Patience and perseverence--I wish you both.
