First, I'll try to define that "internal harmony" requirement a bit, so I can use it to build the faction motivations:
Internal Harmony: The angels are only able to uplift organisms with the appropriate "harmony". This may include extreme selflessness (willing to let oneself be subsumed by a higher power, for the sake of a greater good), or extreme faith (truly believing in the righteousness of the Source, without a hint of subconscious doubt).
Disharmony: The higher realms would all be harmonious, but of increasing complexity. Conversely, the lowest realms (our universe, for example), would be chaotic messes. Our universe is brutally hostile to all forms of organic life. So, only a particularly lucky situation would result in complex organisms...and only the most savagely cunning species would survive to evolve sapience. From the Angels' perspective, humans would embody all of the things an angel cannot be.
Don't Drink the Local Tap-Water: Watchers would be programmed with the same "internal harmony" of the Angels, as the Angels would not be able to comprehend doing otherwise (they couldn't intentionally program something which could disobey orders from On High).
As such, the Watchers wouldn't be explicitly forbidden from interbreeding with humans, because the Angels (and therefore the Watchers) have no concept of sexual attraction to humans. From the Angels' perspective, humans are--at best--psychopathic children. At worst, humans are barely-sentient bacteria. Watchers would initially view the idea of breeding with humans as being on par with bestiality, robbing the cradle, or taking advantage of the mentally ill.
The First Sip of Tequila: The problem is, the Watchers possess biologically human brains, as those are the organs necessary to live down in the gutter-reality of our universe. 1008
Only an organ so chaotic can function in The Disharmony.
Eventually, some of them would start to experience sexual attractions, either instinctively, or as learned behavior of watching humanity for countless years/decades/centuries/millenia. Without a specific order against interbreeding, some watchers might decide to interbreed solely out of temptation, or curiosity, or even out of a genuine desire to "think like a human" (as understanding humanity is their mission).
The Hangover: After a few of the Seraphim (human/Watcher offspring) begin to exhibit superhuman abilities, the Angels would immediately attempt to ban all further interbreeding (implementing a hot patch on all current Watchers?) to avoid contaminating the experiment. Unfortunately, the Watchers themselves would be too disharmonious at this point, so the ban wouldn't implement reliably: free will would get in the way.
Why would the Seraphim have superhuman abilities? The Watchers might be biologically human, but they were designed to be able to communicate with Angels who exist in a higher realm. They'd also be designed to lack any human frailties (old age, susceptibility to disease, etc). Their superhuman knowledge might also be genetic, which means that some of that information could be passed on to their offspring.
Seraphim would be born with some or all of those capabilities, but stunted or warped. For instance, the "Angel Sense" might not be properly attuned to the Angel's realm, and would instead give the <PERSON> a view of spacetime's fourth dimension-- seeing the future or the past (or both). They may even be able to hear other people's thoughts, or project their own thoughts into others' minds.
Physically, a Seraphim might possess almost perfect regenerative capabilities, but still be susceptible to injury. The combination might allow them to constantly become stronger and more resilient over time (muscles growing stronger with use, bones breaking and healing more durably), until they're several centuries old, with a superhumanly powerful physique.
Putting the Genie back in the Lamp: Once the first few Seraphim are born, the human species would be irrevocably changed. Once that Angel DNA is introduced to the gene pool, the superhuman traits would keep manifesting even if all the Watchers were recalled or destroyed.
Over many generations, the abilities would become diluted with baseline human genes, resulting in people with very subtle superhuman abilities: prophetic dreams, uncanny empathy, Olympian strength and endurance, photographic memory, or almost instinctive knowledge of science or mathematics.
Occasionally, an Atavism would occur: a throwback to Seraphim capabilities, or even an almost pure-Watcher (the result of two Seraphim mating, perhaps).
The original Seraphim may also still roam the world, millenia after the Watchers have left, concealing powers which might have grown even stronger over time.
EDIT: Burning Down the House: I'd forgotten to address the part of the original question regarding the high energy requirements of the superhuman abilities. 1008