What is the name of this cast metal part inside my lockset?
From surfing a bit, it seems Baldwin has a variety of handlesets that work with a limited number of their locksets.  If you can't buy just the plate, you might only need to buy the lockset. 

It seems to be called a 'Landing Plate'




  http://www.electronicsweekly.com/made-by-monkeys/materialsmaterials-processing/view-image-youll-2009-01/
  
  We’ve had two Baldwin handle sets fail on our new front door in the
  past six months, and I finally figured out why the manfacturer
  included a second landing plate with the set.
  
  Basically they must have known it would fail prematurely.
  
  The thumb latch is a lever that sits on a set of little pins set on a
  brass die casting, shown above, which allows it to toggle up and down.
  Unfortunately, brass die castings are about making things fast, and
  such parts typically have crappy structural performance and are prone
  to stress cracking.
  
  What probably happened is that a tiny crack formed on one side of the
  casting and propagated every time we squeezed up and down on the
  handle.  One side failed (red arrow) and our door handle starting
  twisting at an odd angle. Thankfully we replaced it with the second
  landing plate before we were locked out of the house.
  
  After going through two sets, we bought a new set and discovered that
  designers finally figured out they needed to reinforce the casting for
  the load by reinforcing the area where the pin sets (right image).