I managed to get it changed, but I fully believe that it was a SW bug or something that was preventing it from updating when it should have...not a procedural problem on my part.
I confirmed that the view in question was derived from the new part (10748). I tried creating a new cut list based off of that view, but that was still giving me the old project number (10583).
So this morning, I tried creating a new view and a new cut list based on the new view. If that worked, I would just replace the old view and cut list...not ideal but easy enough to do.
The new view and new cut worked, it had the correct project number (cut list item names).
However, to my complete surprise, as soon as I made the new cut list (based on the new view), the old cut list finally updated to show the correct project number (cut list item names). By all reasoning, it should have updated initially, or at some point after rebuilding/saving/shutting down etc.....but it didn't. I feel like I've aged 5 years in the 5 months since my company switched to Solidworks.