A cello has an incredibly wide range. The general consensus is about 4 octaves from C2 to C6. This goes from the C two ledger lines below the bass clef staff to the C two ledger lines above the treble clef staff. It pretty much covers all vocal ranges. Now whether or not a given cellist can play the extreme upper register with skill, good tone and good intonation is another issue because the extreme upper register is what is most difficult to play. Most cellists of reasonable ability can at least play into the mid to upper treble clef inside the staff.
Now to address your question of how to reasonably deal with this, Logic will automatically transpose written material to the clef selected so the two things you mentioned that happened:
The accidental for the key signature changing is normal because B in bass clef is one line lower than in treble clef.
The notes did not transpose 2 octaves up but instead were represented in their actual location in bass clef based on the pitch of those same notes in treble clef. I assume there were lots of ledger lines.
When a cello is written in the upper register alternate clefs are used. The first option is tenor clef, where middle C is the second line from the top of the staff. The second option is using treble clef untransposed. Cellists are accustomed to reading all three clefs.
Here is an example (created in Logic) of the first 4 C’s on the cello in all three clefs for reference:
Changing clefs is easily accomplished in the score editor window of Logic by either using either the clefs in the part box (for temporary changes) or by opening the staff style window and selecting the desired clef on the left side of the chosen style.
You mentioned in comments possibly having to transpose the part down an octave for cello. You didn’t provide a reference but if the part is very high in the treble clef and beyond your cellist’s ability and you don’t mind it being played down an octave that is a good option. Still, use a clef that makes the part as easy to read as possible.