"For now, the main point for philosophical discussion is that pleasure is a feeling, and happiness would be desirable because it is a type of pleasurable feeling, otherwise happiness would not be desirable."
To extend this further, it clearly incorrect to say "happiness is not a subset of pleasure, but happiness is desirable in and of itself," because that would create the logical dilemma of there being something other than pleasure which is desirable. That conclusion cannot be true because it is ruled out by the foundational premise that Nature gives us only two feelings by which to choose and avoid, pleasure and pain, of which only the feeling of pleasure is desirable in and of itself.
To say that happiness is not pleasure but is desirable in and of itself would be no different than making that claim about wisdom or friendship or courage or prudence or anything else. All of those are desirable only because (and if) they bring pleasure.
There is no doubt here but that we are working with logical constructions, and that we have to deal with all the hazards that that entails. Some might say that we are playing word games. But that is where DeWitt's observation that Epicurus is the ultimate anti-Platonist comes into play. We can choose to ignore the Platonist logicians and tell them just to go jump in a lake, or we can beat them at their own game and show them that logic can be used to identify accurately the goal of life (as opposed to gods, or virtue, ideals, or rationality itself). It seems to me that since he was teacher in Athens surrounded by logicians of all stripes, Epicurus felt it prudent to equip his students with logical arguments with which to fight off the Platonists. And that reasoning seems to me to be compelling today, since so many people find so many reasons to shrink back from the word "pleasure."
So this appears to be the path in which Epicurus used logic to establish that the feeling of pleasure, rather than an abstraction such as Aristotle tried to do with "happiness/eudaimonia," is the correct answer and antidote to gods/virtue/idealism/rationalism.