Option two will be the best, in my opinion.

Even though the golden gun is famously own for option one, as in, a detective can randomly shoot another random player and hope they kill a T. But this will only cause more problems for everyone. As a new player to the server, you don't expect to get randomly gunned down by a detective and feel they can get away with RDM because they were not punished. As a new player, you expect to be welcomed and to enjoy the server you just joined, not get RDMed by random detectives with an excuse like "Golden Gun".
Option one will only cause more problems then we need.

Option two will be the best. This option will consist of the current rules we have now, this will be the best for staff and users. Once a detective has a golden gun, they can only kill someone for a valid reason. Just as the rules are now, the only difference is that if the detective shoots an innocent, they will die. Adding a new rule just for one detective change will be too complicated and complex, meaning we as staff, will have to inform users that there is now a change in the rules and if you don't get used to it now, you will be punished. Overall, this option will be the best because it is the same as the current rules.
Option two will be the best.

Option three is pretty much the same as option one. People will only use this as an excuse to RDM, and as you said, people just randomly but suspicion on people for absolutely no reason. People doing this will give the detective a reason to use their golden gun and randomly kill traitors.
Option three will again, only cause more problems.

Option four should not happen. Adding a new detective weapon will only benefit the server in every way.
Option four, no.

Option five is no. But adding other weapons is still a good idea.
Option five. Please suggest things.

I'm sure this whole thread will help the Lead Team come to a good decision, so thank you Nay for making it.