I think you're looking at this discussion from the wrong angle. The question is, "how do we differentiate first-order wants that trump second-order wants from second-order wants that trump first-order wants?" Here, the order only refers to the psychological location of the desire: to use Freudian terms, the first order desires originate in the id and the second order desires originate in the superego.

In general, that is a complicated and difficult question, which needs to be answered by careful deliberation- the ego weighing the very different desires and deciding how to best satisfy their combination. (That is, I agree with PhilGoetz that there is no easy way to distinguish between them, but I think this is proper, not bothersome.)

Some cases are easier than others- in the case of Sally, who wants to commit suicide but wants to not want to commit suicide, I would generally recommend methods of effective treatment for suicidal tendencies, not the alternative. But you should be able to recognize that the decision could be difficult, at least for some alteration of the parameters, and is the alteration is significant enough it could swing the other way.

There is also another factor which clouds the analysis, which is that the ego has to weigh the costs of altering, suppressing, or foregoing one of the desires. It could be that Larry has a twin brother, Harry, who is not homosexual, and that Harry is genuinely happier that Larry is, and that Larry would genuinely prefer being Harry to being himself; he's not mistaken about his second-order want.

However, the plan to be (or pretend to be) straight is much more costly and less likely to succeed than the plan to stop wanting to be straight, and that difference in costs might be high enough to determine the ego's decision. Again, it should be possible to imagine realistic cases in which the decision would swing the other way. (Related.)

It's also worth considering how much one wants to engage in sour grapes thinking- much of modern moral intuitions about homosexuality seem rooted in the difficulty of changing it. (Note Alicorn's response. Given that homosexuality is immutable, then plans to change homosexuals are unlikely to succeed, and they might as well make the best of their situation. But I hope it's clear that, at its root, this is a statement about engineering reality, not moral principles- if there were a pill that converted homosexuals to heterosexuals, then the question of how society treats homosexuals would actually be different, and if Larry asked you to help him make the decision of whether or not to take the pill, I'm sure you could think of some things to write in the "pro" column for "take the pill" and in the "con" column for "don't take the pill."

Why I said this is worth considering is that, as should be unsurprising, two wants conflict. Often, we don't expect the engineering reality to change. Male homosexuality is likely to be immutable for the lifetimes of the ones that are currently alive, and it's more emotionally satisfying to declare that homosexual desires don't conflict with important goals than reflect on the tradeoffs that homosexuals face that heterosexuals don't. Doing so, however, requires a sort of willful blindness, which may or may not be worth the reward gained by engaging in it.