Well, apparently I got it to work. My steps were:
1. Uninstalled GTA V via Steam.
2. Deleted GTA V folder in ...steamapps/common/
3. Deleted GTA V folder in This PC/Documents/Rockstar Games/
4. Reinstalled unmodded GTA V.
5. Reinstalled ScriptHookV and ScriptHookVDotNet
6. Manually installed the latest version of LSPDFR
7. Performed first time RPH settings. (Changed time from 10000 ms to 60000 ms. Check marked LSPDFR in Plugins tab.)
8. After first failed start after selecting "Save and Start Game," I started the game, then Alt-Tabbed and started RPH, then reopened the game and started correctly upon selecting Story Mode.
I'm not saying this will work for everyone, but it worked for me. Obviously, I had to start a new game, so as soon as I can get past the prologue I'll update whether it worked completely or not.
Update: Same exact problem even after a clean install. Not sure what to do now. The god of mods is not with me today I guess...