DSDs do not belong under the trans umbrella (though that is what the letter I in the acronym stands for) but the sex category in sport should be about male and female bodies. A male with a DSD is male. A male with an outwardly feminine phenotype.
I question whether a chromosomal test on a teenage CAIS athlete which revealed their condition would really come as a shock. Athletes typically break into elite sport in their late teens at the earliest. Even if we are talking about athletes from poorer countries without the means to do sophisticated sex testing of babies, a 17- or 18-year-old "girl" who has never had a period or gone through puberty would have some inkling that something was up. But let's say that it is a shock, that is not the concern of regulatory bodies. Their job is to promote the values of sport - safety, fairness and inclusion. If we are expected to not bat an eye at the sight of very masculine-looking "women" (who are not women, they are males with DSDs) in women's events, why can't we accept feminine-looking men (who actually are male) in men's events? Of course they would be very unlikely to make it to a high level because of their DSD, but that is the case for many male athletes who suffer from debilitating conditions.
I about taking puberty as the dividing line for several reasons. It's hard to judge exactly when puberty starts, and boys benefit from two mini-testosterone surges before the main event which give them a significant advantage over girls. In practice, a boy whose puberty had been arrested before Tanner stage 2 would probably not be healthy enough to have any sort of athletic career, but you never know what advances in treatment might be available in the future to mitigate the problems of lack of testosterone. Every DSD is sex-specific, therefore it is easy to know who to exclude from the female category. The principle of female sport for females only is important.
Going back to DSD athletes, CAIS women obviously cannot benefit from the supercharging effect of male puberty but they still don't have the disadvantages of female puberty. They can train the same way every day, training does not have to adapted to fit around their hormonal cycle. They will never have a pregnancy scare before a major competition, or suffer crippling pain during it. While not as powerful as men without CAIS, they grow taller than women and greater height is an advantage in many sports. Those are just some examples, and the effect is cumulative. CAIS women are very significantly over-represented at high levels of women's sport therefore there is a strong presumption of advantage.
Unfair advantages are deemed to be so either because of their source or their magnitude or both. Being heavier gives you a big advantage in rowing, weightlifting and boxing, so there are weight categories in all these sports. Height is an advantage but not so significant that we need special categories. All the tall people will have an advantage in basketball or netball, but you still get 5-foot 6 women in high level netball. Being male is the most significant advantage of all. So a CAIS male should be excluded because the source of the advantage/lack of female disadvantage is that they are male. If these athletes were female, with a female-specific DSD which gave them a similar advantage, that might be different.
Categories are created to exclude athletes as a class - not to exclude individuals. Males are excluded from the female category, therefore weaker males are as ineligible as stronger ones.
Being CAIS is a personal matter for the individual. They may embrace their condition or it may cause them sorrow. But it is their burden, not that of their competitors and it is oppressive to make it so. Is it fair on the women competitors to make them compete with an athlete who has natural performance advantages because they are male? I say no.