How to deal with lath and plaster over top plate of partition wall?
Like you're guessing, I would cut the lath as close to the partition wall as possible, then put furring strips [if necessary, see comments below] and drywall over it.  

To cut the lath, I wouldn't use a reciprocating saw except maybe with a very fine-toothed blade.  Cutting those lathing strips with a reciprocating action is going to flex them back and forth.  That vibration may propagate into the other room, causing the lath to pull away from the plaster from above.  The result would be cracks in the wall/ceiling or falling plaster.  

Instead, I would use a circular saw.  Or maybe a circular saw attachment to an angle grinder if you can't hold your regular circular saw up in the air or against the ceiling lath.