How can I use a 120v generator to power both legs of my electrical panel?
Yes! That is correct.  Well done with  the MWBC investigation.
The cheap/fast answer is simple: Since you are familiar with MWBC, it is safe for you to rearrange your loads to place all your critical loads on one phase. For instance, that would mean powering only 1/2 of any MWBC.  Attach the 2-pole generator breaker in  the usual way, but only connect your &quot;hot&quot; to one leg.
For the sole purpose of researching which leg a load is on, it is OK to temporarily pull 1 wire off a 2-pole MWBC breaker, or defeat the handle-ties.  Just put it back when you're done :)
In fact make sure they are handle-tied while you are in there.