I can think of one experiment that would be entirely legal. It would be expensive though. If under age sex causes long lasting emotional damage then those who have been “abused” should be easy to discover. If under age sexual abuse happens as often as it is said to there should be plenty in any small sample. The prevalence being 1 in 4 girls and 1 in 6 boys.
Have adults be evaluated by shrinks. Have those shrinks try and determine who had sexual contact with an adult as a child based on emotional damage. Have those same adults answer a question later posed positively and negatively on whether they had sexual encounters with an adult as a child.
If the shrinks can come close to identifying the “abused” from those who where not “abused” then the claim of emotional damage has merit. If the shrinks can't sort them out then the claim of emotional damage is bunk.