Whether to power them with beamed energy, solar energy, solar-wind energy, isotope energy, or on-board fission or fusion power generators is an example of a question I leave to future generations. They should need considerable power for communication at least, plus very sensitive reception equipment, because they would be communicating partly with tiny craft that could not carry giant antennae to capture faint signals.
Whether there would be relatively few multi-function satellites or relatively many specialist function satellites, and whether a satellite that outlives its fuel supply would be parked and catalogued, or scrapped, refurbished, recycled, or destroyed, I also leave to future interests.
Personally I like the idea of large-scale, modular satellites to be serviced and upgraded by specialist unmanned craft, but I do not insist upon that.
They would rely on the permanent Relay Satellites for most of such functions, and partly for keeping track of the prospecting craft. Of course, each satellite would have its own intelligence and a very large memory, probably petabytes rather than terabytes, enough not only to manage the data that it accumulates, but the parameters of the infrastructural system as well. They would have sufficient intelligence for routine tasks, including some fairly complex ones, because apart from the question of how far robotics would have advanced in the next century or two, such tasks would be, if not highly stereotyped, at least confined to a small universe of discourse. They also would have great redundancy of function and capacity to ensure resilience in the face of predictable radiation and unpredictable accident. None of your single-drive hard disks and the like!
Bodies that either pose a threat to the inner solar system, or that seem to be potentially valuable to the main project, probably would be visited physically to obtain all relevant information. For example, bodies that are rich in ice or ammonia might either be particularly valuable or not usable for the purposes of the project. Similarly, bodies that amount to aggregations of gravel might be valuable if their trajectories were particularly suitable for gentle manoeuvring, but hopelessly dangerous to use otherwise unless they could be cemented, say by combination with an iceberg or ammonia-berg.
All the other craft would require updating, modification, refuelling, repair, and possibly even retrieval. The Relay Satellites might well require being transported to their stations as well.
No one tug or maintenance craft would be suitable for all such functions. However, each one probably would be versatile and each one would have powerful thrusters of appropriate kinds. However, they might need less fuel than the prospectors, because they would have shorter missions, and more closely defined.
We are after all speaking of Rockrider craft undertaking voyages lasting decades at least, and commonly centuries. Even if we condemned convicts to such voyages, it is hard to imagine what we would want them to do out there in space, even if we could trust them there.
I leave such distasteful speculations to readers with the appropriate distastes.
For example, if an otherwise suitable 100 gigatonne object were spinning at a rate of several hertz, the very task of de-spinning it strikes me as discouraging; I would rather go on to look for something friendlier. Nor would we be interested in 10-tonne or peta-tonne objects, or at least that is what I assume. Again, we would prefer to deal with objects whose orbit we could adjust most economically in terms of energy and time. Exactly which variables would be most important in a given case, I do not much speculate upon.
Possibly one could use nuclear explosions for crude preliminary adjustment of some kinds of orbits of suitable bodies, or even to persuade some bodies to collide usefully.
The computing load would be heavy, but routine. Much of it would be done Earthside many years in advance. There would be plenty of time to seek out the most obscure scenarios for each Kuiper Belt object, where each major improvement in handling a single rock would be worth billions of dollars, ignoring inflation.
We would deliver many times more energy and momentum than we had invested. It would achieve either excavation on the target or adjustment of rotation as required.
Together with adjustments calculated in the light of what it finds on the way down, it carefully places a multi-megatonne nuclear bomb ordered in advance in the light of the prospecting report decades before, covers the hole nicely like a cat, retreats to a safe distance a few hundred kilometres away in space and on the sheltered side of the rock, and when the attitude and position are right, it blasts a tidy slice off one end of the rock.
The resultant vector of the blast both kills the rotation or very nearly, and accelerates the rock into an improved, more elliptical, trajectory. You see, the depth of the bomb was such that several thousand tonnes of material were blasted off at a modest velocity, imparting a really efficient delta-V in the desired direction.
In principle such electret propulsion could be far more efficient than ion thrusters. The reaction mass is cheap. In navigating a teratonne rock we could afford to use thousands of tonnes as charged reaction mass without serious regret.
But all is not lost.
Though the ultimate bull market benefits are far in the future, there is plenty of bull for the developers en route. Whole dynasties of companies could profit hugely from running the projects, improving the technology, applying the information gained much as we have profited immeasurably from satellite communications, weather observations, and Earth science and mapping, even while moaning bitterly and persistently about the costs of space technology and research.