It seems the issue was with MIDI communication! I contacted Nektar support, who responded quickly and referred me to this page:
https://nektartech.com/midi-communication-issues/
Following instructions there I downloaded "MIDI Monitor" and saw quickly that the the P1 wasn't transmitting MIDI info. I power-cycled the unit and saw the pattern hold up: 1-2 seconds of successful communication, then a freeze. I have the P1 plugged into a powered USB hub, so I tried plugging it directly into a laptop USB port and the problem vanished. I then went and tested more USB slots on the hub and frustratingly found several more bad ports before finding one that finally transmitted MIDI information without freezing (so far - fingers crossed).
I'm a bit puzzled by these findings. All the ports I identified as "bad" are ports I've successfully used with USB mics, cameras, hard drives, etc. without issue in the past! I don't quite understand why MIDI info would be different. Any explanations would be very welcome.
In summary, I solved my P1 Reaper installation and setup issues by:
1) Ensuring MIDI In/Out was set to ReWire Host in the Control Surface Settings, which enabled the unit to recognize Reaper and leave internal mode.
2) Using MIDI Monitor to check for successful MIDI communication between the P1 and my computer. When it didn't exist, I tried a different USB port and solved my issue.
I hope this ends up being helpful to someone else!