Cosmic Trigger I have never encountered enlightened hedonism before, could you speak a bit more about it?
I can think of at least two ways to interpret what you have said so far:
i) Enlightened hedonism is a form of indirect consequentialism, in that it advocates living in accordance with certain rules or principles over constant deliberation and says that acting in accordance with what brings pleasure to oneself generally promotes good consequences.
This account is not totally implausible, however, it seems to me that in order to be consistent with promoting the best consequences it must presuppose that people are generally much more altruistic than many would think is plausible. It sounds like maybe the self-interested action is one that someone would perform after an idealised kind of self-reflection, in which case the doctrine would turn into a form direct consequentialism. The reason I think the proponent of such a view is committed to indirect consequentialism is because on the direct version there would be many cases in which deliberation showed the way to promote the best consequences was to completely disregard ones own interests, in which case the egoistic component vanishes.
If the view was a form of satisficing consequentialism then the egoistic component might hold up a little better. Satisficing consequentialism says we don't have to maximise good consequences, we only have to promote a certain amount of goodness which satisfies the minimum requirement then we can do what we want so long as we don't harm others. I don't think many people subscribe to satisficing consequentialism these days, for one thing, no matter what level of goodness you decide is sufficient it seems that the decision would ultimately be based on arbitrary considerations which are hard to justify.
There is a problem with indirect consequentialism more generally which I won't go into unless you state you are committed to this view.
ii) The agent should act in order to bring the most pleasure to herself because doing so benefits others and
I find this to be a self-defeating claim. If the reason egoistic behaviour is justified is that it promotes some external good (i.e. benefit to others) then it seems what is really valued is the external good. In this case, if more external good could be achieved by acting contrary to what brings the agent pleasure, which seems quite plausible in many circumstances, then more of what is valued on this account would often be achieved by acting contrary to what is prescribed.
Perhaps we could weaken this account to make it more plausible:
iia) It is permissible for the agent to act in order to bring the most pleasure to herself because doing so benefits others.
This would be better, but ultimately it still looks like what is valued is benefit to other people. It isn't clear why the proponent of this view would want to endorse pleasure seeking behaviour over other kinds of behaviour which benefit others. I doubt if such a view could really be described as egoistic hedonism. Perhaps if the benefit for others was defined in terms of pleasure it could be called hedonistic, but in any case, this seems to be a departure from what you have said so far.
I should say that the reason I formulated ii) and iia) as the agent acting to bring the most pleasure to themselves is because it seems to me this account is most consistent with egoistic hedonism, also because many (perhaps even most) ethical theories don't expressly prohibit the agent from seeking pleasure and I felt this was the easiest (perhaps the only) way to make it a standalone normative doctrine, as opposed to a merely permissible option which is consistent with many more informative ethical theories.