Though my friend's original conception would be that the energy is
recycled by souls of the dead (I object it and say that it draws
energy from ambient energy of the surrounding, say the sun, or
geothermal, or vacuum energy) - Hekdrik Lie
What if you could have both? What if it could be recycled by the souls of the dead, and simultaneously be drawing energy from the surroundings? It turns out combining the two is not as unreasonable as they may first appear, but we need to change the focus slightly. Your question focused on energy, but I would recommend focusing on information instead. The two are closely related, but the bridges you may want to cross are easier crossed with information theory rather than energy balances.
Consider which is more powerful, access to 4MJ of energy (the average energy in a daily intake of food), or the knowledge of where to find a bulldozer with a full tank of gas. Clearly a being with such knowledge could do spectacular feats, especially if nobody else could see the bulldozer for one reason or another.
Maxwell's Daemon is one way of looking at the connection between information and energy, and it is an important one to the world of physics. Basically, the argument is that, once a particle reaches a uniform distribution in a container with two rooms, there is no way to shut a door between them to trap the particle in one chosen room more than 50% of the time. This is considered an absolute rule by current physics. QM bends it, but even QM doesn't break this rule.
But there's a key assumption to the Daemon: the assumption that the particle is randomly distributed at the start of the experiment. What if it wasn't? What if there was structure to the way the particle bounces, which we had simply forgotten. When we go measure it, we see it as a random distribution. But what if the Daemon knew better? What if the Daemon could predict the position of the particle at some time and simply wait for it to get there.
In theory the Daemon could create any steam engine it pleased with this approach. It just may take a lot of dedication. With oracle like knowledge (knowledge not explainable by physics), it could even generate infinite energy. But we're not looking at that extreme, we're looking for something physically realizable.
Landuaer's principle gives us some physically realizable ways to do this. It states that the minimum amount of energy needed to erase one bit of information is kT ln(2) (technicality: The Daemon is erasing one bit of entropy in the universe. This can also be thought of in layman's terms as adding one bit of order to the universe). The hotter you get, the harder it is to erase a bit. In theory, computation done at absolute zero could calculate anything for free, but in reality, there's no degrees of freedom at absolute zero to do calculation with, so we have to just settle for really-really-cold.
A note here: this limit is really really low. At room temperature, it takes a mere 2.85pJ of energy to erase 1 bit of entropy. Our modern computers are a million times less efficient
So what if we could do our calculations in the ultra-cold of space, we could work at 28K instead of 280K (which is room temperature). That gives us an even lower bound of 0.285pJ/bit. That's not much.
Now before we return from Wikipedia and start really tackling the problem, its worth noting that reversible processes can exceed this limit, because reversible processes are theoretically not bound by thermodynamics the same way irreversible processes are. This is the basis for Quantum Computing, but there is, as always a limit. One has to get the data out of the quantum computer, which involves erasing bits. However, you only have to erase enough bits to get the answer out, which is often many fewer bits than you would have needed to solve the problem classically.
And this is where your computronium comes into play: it can do things like this. It can run reversible calculations to emit only the minimum output. In fact, it can emit output undetectably to science, simply by concentrating its classical outputs in locations science is not looking. As long as it outwits the science, it can stay hidden (and note, the shift to terminology describing intelligence is no mistake).
The question is how does it get this information, to avoid being detected. It literally needs to find bits of entropy that are "forgotten," i.e. unusable by science, capture those bits, and ensure science never sees them again. It can't do this with things like solar energy directly - it doesn't have much better of a chance at predicting information in solar photons than science does. However, there is a corner where science doesn't look: death.
Science really has no idea what happens at death. We've got some ideas, but generally speaking, lots of information in the body decays faster than we can even consider looking at it. Thus it is not unreasonable to assume that there is some information in or near the soul that escapes science's net in the form of information that is too ephemeral to really pin down with statistics. This information could be captured by the computronium without wasting too much energy. This information could be used to better predict when and where the scientists are looking, keeping the entity hidden.
This entity could then use this information at its maximum effect. For example, if a fireball was desired, it may be able to summon up an information net across the entire world to create a fireball in one location counterbalanced by a millionth of a degree decrease in temperature around the world. It could then take advantage of the fact that it's the only one who knows that millionth of a degree shift occurred to harness the suns energy across the globe to pay the energy price for the fireball. It simply has to know enough about how the world world works to do that process "reversibly."
And that is the limit of this creature that you need to prevent limitless power. It can only exert its infinite will on small things. The larger a task, the more likely it is that the imperfection in how it does reversible computation will crystallize unexpectedly, like whitecaps on an otherwise energy-conserving sea. The entity has to be careful: what are whitecaps to it could be catastrophically powerful waves of force for the denizens of earth.
As for your negative mana resistance? Now its easy because, instead of being a very strict physical law you look for, it's more of a social law. The entity will exert its power where it is most efficient. If you are willing to look the other way, it can do great things, simply because it can get away with doing them affordable. In fact, if you develop enough of a report with the entity, sharing a common language, it may find it in its interests to actually give you energy. It'd be a win-win. You get energy, and it gets a voice which will effectively evangelize the best ways to not look at it, so it can continue to grow.
There are also really interesting religious subtexts here as well. The most obvious is the Taoists, who seek to become Immortal by becoming one with the chaos of the cosmos. Perhaps the Taoists were one such group which attained favor with the entity, and were given protection in return for cultivating a life which brought more information to the entity.
At this point, the physics the scientist sees can be anything you please. The entity would present itself in a form the scientist would understand. The scientist would understand laws of nature, so it would present as having rules. However, as science tries to tap this "limitless" power, the entity would have to shift to avoid loopholes, slowly teaching the scientists how to live in harmony with it.