What is the name of the part in the floor pump chuck that depresses the valve stem?
Is it standard enough that I could buy a replacement?
I mainly use this pump to inflate presta tires. From what I can tell, the problem is that the Schrader side collapses and blocks the airflow. If I cut off part of the "flap" on the Schrader side, would that likely allow enough airflow for the pump to work in presta mode?
Different manufacturers will call it different things, and there's nothing standard about it at all - many pumps don't even have one part that depresses both valve types. That looks similar to a part on my Road Morph portable pump, in which you have to rearrange the head to fit different valve types. Full size floor pumps generally have a different (bulkier) way of switching between valves.
For some pumps there are service kits available, but it's more likely you'll have to replace the entire head/chuck.
So your options are to buy a replacement head in the first place, or to try modifying what you've got knowing that replacement heads are available.
Agreed with Chris's answer that there is nothing standard about the part: the diameters and lengths will vary by manufacturer, pump model and year of manufacture.
As a technician, I was surprised to find rebuild kits available (I have no affiliation to this link, but it is advertising a similar part for about two dollars, so it can't hurt to ask your local bicycle shop!) for some models of pump chucks: 
You should be able to cut off part of the side that you don't use, but leave two or three millimeters of it there to keep the piece centered in the chuck. It rests on the "shoulder" (the wide part of the bushing), so don't cut any of the other parts.
