Oddly enough, some social trinitarians have argued that perfect being theology supports their view (see Dale's discussion). I shall argue that perfect being theology supports Latin trinitarianism very nicely.
Perfection strongly suggests the presence of perfect interpersonal love.[note 1] Therefore, perfection considerations make it plausible that there is more than one divine person exhibiting perfect interpersonal love. Moreover, love has two kinds of perfections.
The first perfection of love is that of generous giving and receiving—this is the perfection of beneficence. There, the perfection is the greater the greater the gift. The gift of divine existence seems the greatest gift possible. So, at least one divine person has an existence that is a gift of the other, and this person receives this existence gratefully. Note that in generous love, there is no need for any quid pro quo and so the love can be made mutual by gratitude. Moreover, it might be stretching our ability to know about perfection with much confidence, but it is at least plausible that for every pair x and y of divine persons, x and y are related by such giving and receiving, so that either y receives divine existence from x or vice versa (but not both at pain of vicious circularity). Interestingly, this condition is only satisfied by Christian Trinitarianism if the Catholic doctrine of the double procession of the Holy Spirit holds (otherwise, there is no such giving and receiving relation between the the Son and the Holy Spirit).
The second perfection of love is the unitive. All love involves a certain minimal union of mind and will (we try to see things from our beloved's point of view, and we are apt to particularly pursue those goods that our beloved particularly wills), and all love is directed at a union that is more than minimal, a union that is a consummation of the love. The perfection of love will be a consummated love, a love that achieves a perfect union. An absolutely perfect union of love will be one where there is the maximum of unity that still allows for the most perfect kind of love. The most perfect kind of love is interpersonal, so the union of love must maintain a distinction of persons. But, it is plausible (assuming this is at all possible), that everything else but the relations distinguishing the persons, will be in common in the perfect union of love.
In particular, the perfect union of love will, plausibly, involve one mind and one will. Not just in the extended sense in which we talk of two human beings being one mind and one will, but in the literal sense of having in common a mind that is numerically one and a will that is numerically one. (Maybe we can even get the divine simplicity claim that the mind will be identical with the will, but I don't want to insist on that at this point.) This one mind and one will is essentially indivisible, for perfect love will seek an indivisible unity.
We thus get numerically one divine ousia, including numerically one mind and numerically one will, and yet more than one person.
Interestingly, while above I took the route of perfection of union, the route of perfection of generosity can also be used here. Generosity can be perfected in at least two ways. One is with the value of the gift—a better gift is one that perfects generosity more. The other is with the intimacy of the gift-giving—with how close the gift is to the giver. Thus to give money is ceteris paribus less generous than to give an heirloom, and to give an heirloom is ceteris paribus less generous than to give a kidney. The most perfect gift will be one that is both of maximum value and of maximum intimacy on the part of the giver. The divine nature is of maximum value. There are now prima facie two ways for perfect generosity to be exhibited. One way is for the divine giver to make another God, another person with another, qualitatively identical, divine mind and will (or ousia). But this does not exhibit perfect intimacy in the generosity. That intimacy will be perfectly exhibited when the divine mind and will given are the very same divine mind and will that the giver has, when the very same life that the giver has is given to the receiver.
And it is only in a perfectly intimate generosity that reciprocation by gratitude is perfected. For the closer the gift to the heart of the giver, the more the recipient's generosity means to the giver, and when the gift is the giver's own mind and will, the giver's own life to be shared, the generosity can be a deep affirmation of the giver, and indeed is a recognition that one prefers to have one's life as gift than to have it on one's own, and that recognition is a willingness to share that is in principle generous. It is, indeed, a kind of giving back.
Objection: Social trinitarians claim to believe in the unity of God and hence will claim that their doctrine of the Trinity is compatible with everything I said above.
Response: If social trinitarianism is to be distinct as a doctrine from Latin trinitarianism, it, I think, has to claim that Latin trinitarianism posits too much unity in God. But if this claim is made, then the above argument works—for the above account posits the maximum of unity in God apart from the distinction of persons, which is precisely what Latin trinitarianism gives. Of course, if social trinitarianism (which, in general, is kind of hard to define) turns out to be compatible with Latin trinitarianism, then there might be no disagreement here at all.