Although somewhat academic, since pgboucer doesn’t support it (and might not ever), have a look at this ticket which, if it was ever supported, would give you what you needed:
If they are going to keep the client connections open, they would need to run pgbouncer in statement or transaction mode.
As I understand, in pgbouncer you cannot have connections that serve different users. If each individual requires its own database-level user, pgbouncer would not help at all.
I would look seriously into getting rid of the always-open requirement for connections.
