If I point one telescope at a pulsar and record the image and then I point a second telescope at a mirror that has the image of the pulsar on it and record it, will the two recordings be different? It seems to me that according to the theory posted in the link below they should.
I think you're confused about what to consider relativistically here.
The pulsar itself isn't spinning at relativistic speeds. The fastest we've observed has a period of about $2$ $ms$. Even though light is relativistic by definition, the information isn't arriving at relativistic speeds, i.e., the intensity of the light as a function of time doesn't change relativistically. Therefore, if you looked at a pulsar in a mirror, you'd see exactly the same thing as looking at the real pulsar (if you've got a really good mirror), except perhaps delayed by the extra light travel time between the mirror and the lens of the telescope.