Okay, took me a while before i got around to it but I actually checked out what was going on with the module. I found that the tuning was mostly on point, with one or two octaves slightly off, but the first octave was WAY off, playing something around an E when the module had 0V on the input.
Recalibrating the module seemed to fix the issue, but I'm thinking that it may have problems with losing data that should be in non-volatile memory given my issues with the module powering up with the slew rate set to the maximum value. Funnily enough that also happened when I booted it up in calibration mode, so I had to wait a bit for the oscillator to slide up to the new octave whenever I moved on to the next one.
I'm hoping it doesn't do this too much, it'll be annoying having to recalibrate it often. I'll probably reach out to Mosaic as well to see if they're aware of the issue.