Is nuclear power desireable in the long term, given the fact that it's an unnatural heat input to our planet?

There is a balance between heat input and radiation lost to space. To maintain this balance, if you increase the input by 0.1%, you must do the same to the output. Approximating the earth as a blackbody, the energy it radiates is proportional to the fourth power of the temperature. So the temperature would need to increase by 0.025%. That's less than a tenth of a degree, which doesn't seem very significant.

The answer is "No", and here is the qualitative explanation. The Greenhouse Effect is not about extra energy input at the Earth's surface, nor is it about reducing the thermal output. It is about upwelling.

In the static, ocean-free black body model: consider and sensor in deep space looking back at Earth black-body radiating in the infrared: it does not see the surface, it sees 1-optical depth depth down into the atmosphere, and it is that chunk of atmosphere that is at the correct blackbody temperature to balance the energy input. As you penetrate more optical depths into the atmosphere, it gets hotter, down to the surface, where it's hottest.

So the surface is hottest, radiating isotropically, with energy transfer to cooler regions above, which do the same, and so on. The processes are refer to as upwelling and radiative transfer.

The addition of greenhouse gases increase the optical of the atmosphere, thereby raising the surface temperature without changing the energy input or output.

According to this source, the total solar irradiance of Earth varies-- rather conveniently-- by about 0.1% over decade long solar cycles. Since the climate is very difficult to predict accurately, unfortunately I can't find any widespread consensus on how much this affects climate (although I'm no climate scientist so there may very well be one that a quick search isn't producing). However, it seems likely that these solar cycles have been around much longer than man-made climate change has been an issue. So, at a first glance, I would say that a 0.1% increase of heat input to Earth wouldn't be too big an issue.

EDIT: To address the OP's concern that the cycles involved in the solar cycle are too short lived to make a valid prediction, I went down a rabbit hole of climate science and learned about Milankovitch cycles. Essentially, the obliquity of earth's axis, it's precession, and the eccentricity of Earth's orbit around the sun all vary in long (tens of kYr) cycles. However, only eccentricity actually changes the total yearly insolation of Earth-- according to this source, by about 0.167%.

So, now we have a very long-lived cycle of insolation that we can compare with and get (hopefully) more accurate results. That same pdf helpfully compares a short-time Fourier transform of average global temperature to STFT's of other data-- namely obliquity, eccentricity, precession, and insolation at 65 N in July.

In the pdf, they show that the STFT of eccentricity does in fact coincide with that of temperature for certain frequencies. So, one might be tempted to conclude that the 0.167% change in total insolation caused by eccentricity was responsible for the $12^\circ$C fluctuations in average temperature over 800 kYr, which is certainly a worrying amount! However, there are many other factors at play, as obliquity also shows a strong correlation to temperature even though it doesn't affect global insolation, as does insolation at 65N in July. Of course, this all stems from the fact that the Earth's climate is very chaotic and interconnected. For instance, lower insolation at 65N is more likely to produce ice than at lower latitudes, which leads to higher albedo and a whole bunch of other cascading effects.

The takeaway from this is that it's pretty difficult to know conclusively what would happen if humans dumped an additional 0.1% of total heat into the environment without actually doing it. However, there are natural processes that change heat dumped to the environment by comparable amounts, so it's not completely unprecedented (although the time scales involved are obviously very different). I think it's also important to point out that 20 times the energy usage in 2013 is a very extreme estimate, so the real life effects would probably be significantly milder than this post makes them appear.