You are talking about simple superhet, not quadrature mixer? Then LSB and USB will always touch at 0 kHz signal frequency. To select a single side band by filter, you need a super-steep crystal filter. Shifting the IF doesn't change anything.
Yes I am talking about simple superhet, not quadrature. It is better to explain my thought with an example.
Assuming a tone on 10.003MHz mixed with 9MHz LO.
sidebands output= 1.003 (USB) and 0.997 (LSB)
But, if I use a ceramic filter at 1.030MHz instead of a crystal filter at 1.003MHz then I will reject the LSB (0.997) as well as the USB (1.003) and instead of detected audio, I get a 30KHz second IF signal. The signal BW will be larger, due to the ceramic filter used, but small enough to reject the close to carrier sidebands produced in the previous step (both!).
Humans cannot hear to 30KHz, so next step is to downconvert this 30KHz into audible range.
With a LO of say 27KHz, the outputs will be 3KHz and 57KHz.
The 57KHz can now be easily removed with a 3KHz audio LPF.
I have not checked about inversion of the audio spectrum yet due to the mixing actions, but at any case re-inversion can be done easily on the audio domain.