I've nearly finished implementing this, but I've realised a couple of things that don't make sense to me.
For simplicity I refer to the PRs as Kings and the Assassin.
While all 3 PRs remain alive, and the Assassin still has both shots, a King using their kill is an instaloss. No matter what else happens, they'll be shot by the Assassin. The only reason they would use it is to kingmake if they know they're going to lose. So why even allow this option? Wouldn't it be better to only allow a King to kill once another PR has been eliminated?
How important is Twilight? I don't see the need to allow the Kings to use their shots during the "Assassination" phase (that is, the time after the Assassin is killed or exiled when they are able to use their shots), as doing so would simply allow the Assassin to make more-informed shots. Should an exiled King be allowed to make a shot during their own Twilight? If so, they may as well have an Assassination phase of their own in which they can use their shot to turn the tables in the same way an Assassin can. I'm pretty sure this is not the intent.
The game state after an Assassin is killed but misses both shots seems strange. If the teams are equal in size, exiling a King is impossible, although nobody knows whether it's possible. I'm trying to wrap my head around whether there's anything to be done here other than just taking shots based on what has already happened. Is it possible to Kinghunt without doing more harm than good to your own King's (or self's) secrecy? Maybe you can argue that a majority of eliminated Guards seemed to be protecting the one person, and therefore your team is now the majority and should lynch that person. Of course, if you're right and your team gets on board, that person will take a shot before the hammer, so your King needs to decide how likely it is that you're right and how likely it is to result in a correct shot. So maybe there's something there.
If something like this doesn't happen, though, then it comes down to the shots. If the Kings have more than 50% likelihood of hitting their target, then the first to shoot has the advantage (which is not good in standard games but might be okay in Party games?). If it's less than 50%, then both will just wait until deadline and see what happens. So maybe it's better to just have them shoot simultaneously and be done with it?