Another back-of-the-envelope calculation:

Total solar insolation is 42 PW, about half of which is absorbed as heat in the atmosphere and at the surface (and ultimately re-radiated as infrared).

Suppose that an increase by 3% would be sufficient to cause cataclysmic warming (i.e., raising mean temperature from 288K to 298K). Presumably, that would require about 3% more heat.

So our budget is 3%, or 1.26 PW. According to

https://en.wikipedia.org/wiki/World_energy_consumption, the entire year of 2012 used 5.6 x 10^20 J, or 18 TW average. Humanity has about two orders of magnitude to go before running into climate problems by brute force alone (i.e., not counting greenhouse gasses and such).

FWIW, at present exponential growth rate, energy use (fuel, thermal, electric, everything) should take about 250 years, i.e., by about year 2260, to reach that level.

A change of 3% insolation also corresponds to an average 1.5% closer orbit. (The halving comes from the inverse-square law: for small changes in r, the change in power goes as -2 dr / r.)

Suppose we were able to shift our energy demands into pure orbital energy. Suppose we used as much as possible, to maximize power use without absolutely destroying the climate (i.e., maintaining that 3% power gain, steady).

Rather than exponential growth, this would have an inverse exponential curve, because as we use more orbital energy, the Earth moves closer, and more insolation means less margin for direct power use.

The Earth has a potential-kinetic energy of 2.7 x 10^33 J.

The required annual change is initially 1 x 10^19 J. This rate could be held until the orbit decays that 1.5%, that is when 1.5% or 4 x 10^31 J of the potential energy has been used. This would take about 4 x 10^12 years to consume; but because it must decay exponentially, it will actually drop faster than that, at a time constant of 11 x 10^12 years (I think).

Did I do that backwards or anything?

If it truly is on the order of trillions of years, then over the next two billion, we will have to contend with solar instability. Assuming the Earth (and any intelligence on it) survives that long, it may be sufficient to ride out our gravitational potential energy for quite a long time; indeed, once the Sun shrinks to a white dwarf (which itself will give off quite a bit more intensity than it does today, and for a much longer time), and gradually cools off, this would be a fine option, in combination with, say, solar shades for controlling planetary climate.

Tim