How much radiation would something need to output to sufficiently affect the CMB? Let’s say that I had some device capable of outputting insane amounts of radiation, across the entire EM spectrum. If I wanted to affect the CMB to the point that, were Earth-based scientists to know that the device existed, they would be unable to gather any useful data from the CMB regarding the early universe, how much radiation would this theoretical device have to output? Peaking in what frequency? Where would be the optimal place to put it? Or multiple copies of it?

If someone who actually knows stuff about the CMB is scratching their head thinking I’m nuts, I probably am. I don’t know much about the CMB, except that it’s random EM waves that are leftovers from the Big Bang, peaking in the microwave frequency range. So if what I’m describing is absolutely implausible across all frequencies and amplitudes, I would appreciate if you could explain why, given my low level of knowledge on the topic. 

In the event that for certain waves what I’m describing is possible, I would be curious to know if there’s actually anything in the known universe that could do it.
It is not very clear what you mean. There are many cosmic sources of microwave radiation that emit so much energy and/or are so close to us that they confuse measurements of the cosmic microwave background. For example "compact sources" such as giant molecular clouds, supernova remnants, H II regions; "diffuse" sources due to dust in the galaxy and in our solar system. In order to get a clean signal, these foreground sources have to be removed from the data. There are also more subtle cosmological effects caused by lensing and gravitational time dilation caused by the large scale structure in the universe.

There are many ways and techniques that are used to characterise and remove these backgrounds (e.g. see Ade et al. 2013).

The most confusing background source would be one that is diffuse, spread over the whole sky almost uniformly and with a spectrum that is very similar to the cosmic microwave background. The source that most closely matches these properties is dust in our own solar system. An imperfect subtraction of this may have led to controversy over possible large scale anisotropies in the CMB- for example see Schwarz et al. (2015); who discuss the problems of removing the signatures of solar system dust.