This is a good question. You’ve done the hard part already, so if you can validate rational self-interest, the rest follows pretty naturally. We seem to have at this point grasped and validated our standard, which tells us to pursue our rational self-interest, and the question remains “how do we actually attain our rational self-interest?”
When we look at our starting point after we’ve validated that reason can and must direct man’s acts to the attainment of the objective good for man by examination of the nature of ourselves and of things, this is exactly what points us to what we need to look for in a code of generalized virtues. The first thing we need to do is to do exactly what we have said we must do: examine our nature to see what we require in order to be able to find the good.
Well, what is our nature, then? If we were some kind of cyborgs who have to download the good from some database, then our prime virtue would be: go download the good as whenever required. If we were such that meditation in a certain way would allow us to know the good, then our prime virtue would be to meditate in that position. But, as we have already noted at this point, our nature is such that our survival and well-being depend on perceiving and identifying reality correctly, and acting accordingly. Thus, we arrive at rationality as the cardinal virtue. This is pretty obvious, seeing as how we started off by trying to find a rational ethic, so it makes sense that rationality should be the main generalized virtue in the code that will result from this. So we can see that it directly flows from what you have already identified: “His nature demands that he adheres strictly to reason.” Since rationality is our cardinal virtue, all the other generalized virtues will be derivatives of that (independence, integrity, honesty, justice, etc. Productiveness and pride would be exceptions to this, and so obviously come from examining different aspects of our nature.) Once you identify the nature of the good and man’s means for attaining it, the code itself will be an expression of that, spun out into various contexts.
It might be asked at this point why do we need virtues at all, but as per above, we have already noted that we are trying to attain the good for man, i.e. you have already identified our standard of value. Virtues are a requirement, then, because they are the means of gaining values. So, any virtue we do find will have to be judged according to the extent that it advances value-attainment. The proper values are the ends that fulfill and advance man’s life, so virtues are generalized prescriptions for the kinds of actions we should routinely take, and the habits we should cultivate, for achievement of those ends. We have characters, in other words, and we need to cultivate the proper character in order to be someone who can attain the good.
It make then by asked, but why do we need generalizations, principles, or a code specifically? We need to generalize certain patterns of action into principles because reason does not function merely by the range of the moment without integrating the wider context and the long-range effects of actions and character over a whole lifetime. Man’s mind does not work by considering, in every decision, all aspects of the present situation from scratch. We induce generalizations from the fundamental nature of things, and deduce from them applications to specific situations.
That these broad abstractions can be summed up in generalizations called “principles” and applied to specific circumstances to ensure we are in alignment with long-range success. We must integrate the most fundamental facts that we know about our identity and the kind of world we live in into broad fundamental truths, and then apply these principles to the particulars of whatever situations we encounter in the context of our lives. Once we recognize the effects a fact has on us, by applying our standard, we can judge individual actions at any time or place. Since natures are identifiable, the interactions of the various entities will be replicable under the same conditions. The same causes will always yield the same effects. So by examining entities, identifying similar kinds of things, and classifying them into categories each with its own properties and nature, we can arrive at a list of main virtues that form the corpus of our ethical code, that will aid the process of contextual judgment (and that’s basically what Rand does with her list of rudimentary virtues in “The Objectivist Ethics.”)
So on some of those particulars, like “Do I need romantic love, friendship, art, etc.” what we are basically doing is examining the objective effects these things have on man’s nature in pursuant with our standard of advancing what is good for our nature. The code itself is basically just asking “what are the means of attaining life-promoting values?” So we might need some knowledge of psychology for example, or of biology, or whatever (e.g. for the question of how do I know eating this piece of bread is good for me.) Some of these things might be more obvious than others, but if we find that we do have some kind of a need for friendship, then attainment friendship is a good. Then it might bring up a question like “if I want to attain friendship, is honesty effective for this, or is deception better for this?” and so we can then apply the generalizations to these specific instances.
Anyways, hope that helps, but if not, the best book so far on what the actual Objectivist virtues and their derivations is Tara Smith’s Ayn Rand’s Normative Ethics, which includes footnotes to several other Aristotelian philosophers. This is basically Aristotle’s view of virtues as the character traits, or dispositions, which are the means of attaining happiness and flourishing (although she didn’t endorse his Doctrine of the Mean.) There’s also a Peter Schwartz lecture on the ARI website where he touches on this topic. Of course, you can also consult pretty much the whole Chapter 8 of OPAR. And also the great intro book by Bernstein Objectivism In One Lesson has a chapter on “Virtues as a Requirement of Survival.”