It seems your question comes down to how to demodulate a SSB (single sideband) signal.
Note that the single sideband is already the signal you want, just shifted by some fixed frequency. One way to recover the original baseband signal is therefore to frequency shift it. This is usually done by multiplying it by the fixed frequency. Remember that such a product results in the original signal shifted by + and - the fixed frequency. By choosing the right fixed frequency, you slide the SSB signal in frequency-space down to where it becomes the baseband signal. The other part of the result (slid to higher frequency instead of lower) is so much higher than the signal you want that it can be removed with a trivial low pass filter.
You mentioned LO (local oscillator) in your question. That is not relevant here, other than an LO might be used in the front end of your receiver to help filter in only the narrow frequency range of the signal of interest. An LO is part of a superheterodyne receiver, which is really about being a highly selective narrow band filter. In other words, its the part of a radio that tunes in a particular station and tunes out the others. It doesn't have anything to do with demodulating the signal in AM and SSB receivers.
The term for the fixed frequency used to demodulate SSB is "BFO", for beat frequency oscillator.
In this case would the BFO still be a tunable 7MHz - 7.3MHz oscillator since that is the range of carrier wave to be re-inserted? Or is it a totally fixed frequency across the entire spectrum?
You are again confusing the various blocks in a typical radio receiver. It would be very unlikely to apply the BFO directly to the received signal. You would normally have a superheterodyne front end that did the tuning, and converts the selected received signal to a fixed frequency. For SSB, the BFO would then be applied, so in that case the BFO would be a "fixed" frequency.
Some SSB receivers allow for adjustment there to fine-tune the BFO to match the signal. With SSB, even as little at 50 Hz mismatch can make speech sound strange, and with 100 Hz it sounds mostly garbled. That's more accurate than the tuner is usually intended to tune for.