PS4 folks who aren't able to play co-op online, this method worked:
https://gamefaqs.gamespot.com/boards/261123-borderlands-game-of-the-year-edition/77613405/919982664
The beginning of the instructions didn't make sense at first. The poster doesn't mention that you have to be invited by the other player.
So this is what it looks like:
*The intended host will need two controllers and a second PSN account on their PS4 (plus is not required for the second account).
1. The intended host starts a game
2. The intended host joins their own session in split screen with the second user/controller
3. A player who wants to join the session online sends an invite to the intended host
4. The intended host accepts the invite through the PS4's OS while the second user/controller are still in their active lobby
5. The connection to the other player will fail
6. The intended host starts another game, this time just hanging out in the lobby
7. The intended host joins their own lobby with the second user/controller
8. The intended host sends an invite to the online player who's game they just failed to join
9. The other player accepts the invite and should appear in the intended host's lobby
10. The intended host can have the second user/controller quit
11. Start the game
Unfortunately this only works for 2 players co-op. Anyone else who tries to join the game in progress will still time out.