...so maybe I need to buy another NUC at some point. This setup will be fine for now for testing.
Would you be able to stick in one of those cheap wireless mouse transmitters to trick it? If you have an extra Logitech Unifying receiver laying around that could work. Or a $10 wireless jobber like this : Amazon.com: TeckNet Classic 2.4G Portable Optical Wireless Mouse with USB Nano Receiver for Notebook,PC,Laptop,Computer,6 Buttons,30 Months Battery Life,4800 DPI,6 Adjustment Levels: Computers & Accessories - (https://www.amazon.com/gp/product/B00ECZFDYC)
I use the Logitech M585 .. it pairs to one PC via BT and a 2nd via the unifying receiver. Once it installs, you can switch between machines with a button on the mouse or by dragging the cursor to the screen edge. You can also cut and paste between PCs.