I would like to suggest the addition to a multi-server selection option specifically for litebans, since I do not know how the other ban management plugin works. But the easiest way to do this without a bungeecord bridge would be to have an option in the config that you can enable like

# Server Selection Feature
enabled: false
servers: -|
  towny
  survival
  creative
  hub
  global

# Server Selection Feature
enabled: false
servers: 
  - towny
  - survival
  - creative
  - hub
  - global

This will introduce a new GUI section to punishments. When a staff member/moderator runs the "/punish" command it will open a GUI where they will have to select the server they would like to perform the punishment on. After you select the server it will go to the normal punishment menu. If you want you could have the plugin generate a separate file for the server GUI configuration where it will just place the icons in the first available slots and the user can edit the config file however they want (for cosmetic reasons).

This is how litebans handles the bans on bungee.
/ban [-s] <target> [duration] [reason] [server:*]
For example: server:survival to only ban/punish a user on the server named "survival"