Regular restart Freelancer, or even restart Liberty gives you neutral Corsair reputation. Then you have to make missions on Leon since Cadiz missions are broken.
Bar patrons that are the same faction as the station usually do not offer bribes (sometimes they do, but usually not). However the patrons from other factions will usually offer you one or more bribes if you are above 0 rep with their faction (sometimes not, usually they do). Also the bartender is linked to the base, so if you have enough rep with the base owner's faction to dock there then you have enough rep to to talk to the bartender (although it is possible to make them stop talking to you by dropping rep too far negative while you are there).
The zoners in the bar belong to the same faction as the owning faction, so they probably dont offer anything. They should have rumors, but some of the discovery bases are dead like that. However you really want to get bribes from the Corsairs in the bar anyway and not the zoner patrons. If the Corsair patrons aren't saying anything or offering a bribe (all corsairs have an empty chat bubble) then you are most likely negative rep with Corsairs faction, so you will need to get above 0 rep with them first. Alternatively if you are already +6 rep with Corsairs then they wont offer a bribe to you either, since they cannot raise your rep any higher than that anyway. So you need to get your rep between 0 and +2 to get good bribes. The first bribe will take you to +6, the second and subsequent bribes will just add the same amount of reputation on top of that.
Tthe process for a multi-bribe is as follows: dock, see if bartender offers a bribe you want, if not F1 and relog the same character to reload the base data, and repeat as needed until the bartender is offering what you want or you figure out he never will. Once you have that sorted, check the bar NPCs to see if they are also offering bribes you want, if not switch rooms to reload the bar NPCs until you get what you want. Then take the bribes from the bar patron(s) and take the bartender bribe last. I do that every day.