I'd like to issue a little warning here against the use of the word "eugenics". This term is often used to describe not only the goal of transcending human limitations per se, but also certain ineffective and morally despisable means to do so, e.g. the direct killing or sterilization of those that you think have an "inferior" genotype. With such means, the goals you describe would obviously be utterly unattainable.
I understand that the means you are having in mind here are in a completely different class, but I'm a little worried that the explicit debate of "eugenics" at imminst may be misunderstood by critics of the institute. That is why I'm adding this comment.
A less controversial way to express what you mean might be "cybernetics", "extropy" or "transhumanism".
While under most any circumstance reproduction will not present a problem, it will eventually've to be regulated, at least somewhat, IMHO. The resources in any location are finite, and while we can export/import addditional resources eventually that too will hit a limit, and in any case the number of resources that can be kept/accumulated at any location at anyone time are finite also(unless the laws are more flexible than they seem.). Given that there's a limit to that( resources), even within an artificial reality, the number of individuals that can live with a desired flexibility in use of resources within a particular volume is limited too.
AKA, after a point such society will either have to a.) self-limit itself, b.) some members will have to move out c.) said society will have to use power to force some limitations or face less than desired flexibility in use of resources(lower quality of life than desired, however beyond the present day it may still remain.).
As for modification of human nature, to me it's highly desirable. It's my belief that next-gen replicators will most likely borrow heavily, aka will be heavily based, on the biological/organic molecular structure/design/organization/machinery. Given that, I think it may very well be viable to modify the existing natural replicators that compose the body into next-gen ones... but that will require the ability to deliver/edit/change vast quantities of molecular memory... it seems it may even be desirable to edit some in situ(like those harboring the structural information to call/bring/summon-forth the ultimate, the mind.). Delivery/editting mass quantities in situ, and ensuring efficiency in such a task, may very well require a new type of delivery agent/vector, something that goes beyond such definition.
How powerful would that be? Next-gen replicators, as many've stated, can most likely store and transfer vast information about that which they compose/make, the structure they're part of. Such replicators can be made to be very versatil, capable of intelligent modification, and highly-resistant. With such capability, it finally becomes feasible to achieve the dream of man. By taping into the very nature of this world, the nature... of a world made of information, by understanding such, by mastering such, that which many deemed impossible finally becomes possible. Man has learnt across the ages of the nature of information, that by embedding and replicating it, by spreading it efficiently, and by keeping the keys to its continued understanding and replication at hand throughout the centuries, information can trascend. It can be made nigh-perpetual, nigh-everlasting, immortal within a particular reality, concept.
Finally, the world will change... those who're the least fit will be unable to even injure the fittest. Even the weapons powered by the fundamental unit, by the atom itself, will be rendered powerless. Postman need not fear man, for man is powerless before him, for this appears to be the outcome of the laws of the universe, of reality itself. This appears to be the absolute truth written across the very fabric of the cosmos, that those blessed with divine grace, intellect, and beauty, those closer to the divine need not fear, for theirs is the world. The verdict has been written before the dawn of time, logic itself dictates, that man is not the heir of the universe, the princes of the universe, are to be his descendants.
It seems the end of the age of men is at hand, at last, and trascendence is to take place. To the pantheon, to the celestial sphere, to that which has been deemed divine/trascendent/celestial throughout history, before history... will the mind finally arrive, finally its nature is to be unshackled, finally the blessings will be at hand, finally that which processes information will be king of all it surveys... power without end, boundless/infinite/limitless, will be bestowed upon the descendants of man.