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“an interesting idea, here, though - that God might actively interfere with the fusion in the Sun and turn it up or turn it down..”

Another idea I find interesting and, perhaps explains what's going on. Are you ready? Don't flame me and call me a kook!

The sun (and all the other stars) is NOT a nuclear reaction, but instead, an electrical one.

Prior to Einstein's involvement in physics in the mid 20th century, there were many physicists who theorized this might be the case. Tesla was one those men. Prior to Einstein, Physics was a science dominated by objective experimentation of observable phenomena. Much of the research was pointing to the presence of a universal field of energy (ionized particles existing everywhere). Some, like Tesla, believed we would eventually be able to tap into this field for our electrical energy needs.

Einstein, primarily a Mathematician at heart, transformed the study of physics into a mathematical endeavor, with proofs being mathematical rather than experimentally testable. Because of the success of his methodology in the field of atomic energy, Einstein became almost a demigod to both academicians and the public alike.

However, now, even his most famous theory “Relativity” is starting to show inconsistencies with the universe as we've been able to better observe it in the years since his death.
As a result, the mathematical-physicists have had to invent new concepts such as “dark matter” to try to make the theory still work.

Some physicists are returning to the idea of a predominately electric universe theory, and they claim that theory matches up with the observed phenomena perfectly.

How does all this relate to the solar cycles? The field of ionized particles (now observed) in space, while universal, is not of constant density, (much like a visible fog thins and thickens as you move through it). As our star orbits around the galactic center, it passes through varying field density. This field is the source of the sun's energy which behaves more like an electrical node than a nuclear furnace. The sun dims or brightens in accordance with the field density.