MWBC exists, use for bathroom?
Ed Beal's answer covers the core matter.
The handle-tie is mandatory, period.  So that's the end of that inquiry.
You're right that with handle-ties, one will probably trip the other.  However FYI that's no guarantee*.
However, I agree with keeping bathroom lights off the receptacle circuit.  What you might be overlooking is that a) lights and fan are quite small loads (less than 1 amp together)... and b) there are NO circuit restrictions associated with hardwired bathroom loads.  They can be on any circuit**.
Since presumably you have basement lighting already existing, you can simply tap the basement lighting circuit for the hardwired bathroom lights and fan.
The only gotcha I see if that if you have lighting or fan inside a shower stall, it must be GFCI protected.  However basement receptacles are also supposed to be GFCI protected, and you could always place the shower stall light/fan on the GFCI-protected &quot;LOAD&quot; side of a basement recep circuit.

* Factory 2-pole breakers rated for common-trip actually have an unseen internal mechanism which guarantees common trip. Handle-ties are not enough, because of breakers' &quot;trip free&quot; feature: they will trip even if the handle is held/locked in the &quot;on&quot; position.  The handle moving to the tripped position is merely a courtesy indication; if a handle-tie were to tram or bind, the tripping breaker would just trip anyway and fail to trip the other breaker.
** Except for circuits reserved for other stuff, e.g. dedicated circuits required for kitchen, laundry room, garage, and bathroom (though see the 210.11(C)(3) exception NoSparksPlease mentions above).