Correcting large variable dc bias in signal FFT
Based on the plots shown in the OP, it doesn't appear that the signal you've noted as "bad" just has a DC bias. If there was a DC bias, then it would be expressed graphically as a bulk shift by some amount in the complex plane. Compared to the "good" example, it appears that the "bad" one has some time-varying amplitude modulation, which, when viewed as applied to a phasor in the complex plane, could trace out the loopy patterns shown in the plot.

This supports your claim that you tried just subtracting the mean of the signal and saw little improvement; while there does appear to be a nonzero DC value in the "bad" example, the differences aren't isolated to just the mean value, as I said above. The apparent amplitude modulation will add spectral content over some swath of bandwidth of the signal, which could overlap with the region that you want to analyze.