I'm going to suggest three things that might be contributing to the problem you're identifying.
First due to the long course of history, the meaning of arete (translated as "virtue") in Aristotle is often lost in the English term virtue. The term arete refers to excellence.
Second, eudaimonia as you note is not always best rendered by the English word "happiness." (Still, some translators, like Terence Irwin, do translate it as "Happiness" and provide a lengthy explanation of how that's not ideal.). I'm not sure "objective state of fulfillment of purpose" is the best definition for it either (whether we're talking Book I or Book X or the Nicomachean Ethics), but I'd say it's better on a first reading than "happiness" by itself. Instead, I'd suggest seeing eudaimonia as both flourishing in accordance with one's kind and the happiness that results from that fulfilment of the capacities of one's kind.
With these two things in hand, it's helpful to realize that arete is excellence in a particular area of human excellence. So for instance, courage is excellence in managing feelings of fear and knowledge of danger. And this requires knowing both what one can and cannot do and understanding one's feelings. Thus, courage is worth having both because it's ideal as a course of action and because having it contributes to the overall eudaimonia of the self.
These two items might not get to the heart of your question, which my apologies I had a little difficult parsing at 6:50 am local time. It sounds like the core of your question hinges on denying something Aristotle accepts (but that if memory serves the Stoics deny with respect to virtues), namely that something can be both for itself and for a further end.
There's several different things tangled up here. First, I'm going to suggest it's quite plausible to do an action for multiple reasons, but it's quite hard to parse out how much each reason contributes it. So for instance, I might help my brother's business out both because he pays me and because he's my brother and because I get a sense of fulfilment from helping him. If you don't think so (the principle -- not the example), then clearly no account of how one can pursue human arete for itself and for eudaimonia is going to work.
If you can accept at least that one can do an action both for itself and for something else, then the trick for any virtue is that I can want that virtue and want eudaimonia as a result of that virtue. For instance, I run pretty regularly both because I enjoy running and because I want to be in shape and because I see exercising as an achievement. "Wanting to be in shape" also contains a kind of compound purpose since it relates both to the way that being in shape contributes to my general health and to my happiness and to my future health.
Aristotle's claim is that the virtues are just the sort of excellences that mirror and implement well-functioning of the human kind, and that well-functioning of the human kind is eudaimonia (excepting BK X).
Thus, for instance, it's not impossible for me to want friends and to want to be happy by having friends. But to do so not in way where friends become pure utility or pleasure-sources for me (the argument of Nicomachean Ethics BK VIII).
For the Aristotle stuff, some really good resources on this point: J.L. Ackrill, Terence Irwin, and Martha Nussbaum. There's also a volume edited by Amelie Rorty that's very helpful on the question you're asking.
I don't think this account will work as well for the Stoics, but I do less work near the Stoics. If memory serves, their account of how virtues work is such that virtues are not pursued simultaneously with happiness. A key distinction in contemporary philosophy that might help is that they distinguish "right" and "good" and make ethics about the "right" whereas for Aristotle, the "good" has priority over the "right".