Ah okay, after much trial and error, I see what is meant by the issue with antialiasing. I originally thought this was in reference to the top of the model not being flat due to inconsistencies somewhere in the infill, but now I understand you were talking about the very outside edges of the lines/walls.
For anyone following along that may be running into the same issue: Antialiasing was causing a slight inward taper to the 3D shape of the walls. So the bottom slices looked fine, but the taller the walls, the more splotchy the slice layers became as the wall thickness shrunk slightly narrower than it could register. For clarity - this is the part that kept confusing me. I didn't understand why most of the slices on the lower parts of the model were fine, but the top ones seemed to become arbitrarily crummy.
After I removed all the antialiasing and made the image completely solid down to every pixel, the slices were solid and consistent all the way to the top layer, hooray!
For the most part, this fixes my initial problem. I just have to make sure the line art in my drawings make a thick enough wall, and remove all of the antialiasing to maintain consistency all the way to the top of the model.
However, the slices (and printing) were still perceptibly wobbly. The other models I have downloaded and printed, specifically a little crown whose smallest point was set to the exact same width as my logo (.4mm), came out perfectly straight and crisp. For as awesome and convenient as height mapping an imported PNG is, I see there still remains a limit to what you can get out of something made from pixels (especially when you are printing this small).
Time to use vectors I guess!
I imported the PNG into Inkscape, traced and exported it as a vector (.svg), imported the vector into Fusion 360 and modeled it into the 3D stencil, and then finally brought that new model into Cura to slice. This made a significant improvement compared to the Cura height map version (which you can see compared in the attached screenshots)! Unfortunately it's a bit more work than I initially expected, but I've finally achieved the result I wanted.
Thanks muchly for the responses, I honestly expected people wouldn't even bother with my noobishness. I'll mark this as solved, please enjoy a picture of the little cat face logo army I unintentionally created through troubleshooting. 👍