I've been monitoring my own signal with an SDR and notice my SSB transmission extends well beyond the expected 3 kHz bandwidth. An ideal SSB signal shows spectrum of about 70 Hz to 2700 Hz that's rather flat in strength throughout the passband, with no energy outside the passband.

At more than 6 kHz separation the distortion energy level should be at least -40 dB compared to the pass band signal. My signal appears to have significant energy at -30 dB up to 8 kHz from center frequency.

Running a TS-590SG with built-in processor on level 3, external Behringer compressor, and Shure SM58. This type of spectrum usage is not considerate regarding fellow contesters and could be subject to disqualification.

What's the most effective approach to identify and eliminate the source of these spurious emissions? Should I start by bypassing the external processor entirely?

Definitely start by bypassing that external compressor. Compressing audio for SSB introduces harmonic distortion, generating harmonics up the audio band to 6kHz for 2nd harmonics, 9kHz for 3rd. Most external audio processors create more problems than they solve in amateur service.

Your TS-590SG processor on level 3 might already be doing too much work. RF speech processing is superior to traditional AF clipping or AF compression for achieving highest ratio of average-to-peak power. Try the internal processor on level 1 with no external processing first.

That SM58 is part of the problem - it has a very peaky frequency response that's great for vocals but terrible for SSB. You want fair amount of bass for "in the same room" audio quality so it doesn't sound tinny, but also higher frequencies for broadcast quality. Consider a flatter response mic like the EV RE20.