I found a much easier way to do this. To avoid the drop outs and frustration of having to this over and over again you can simply have 5 missions ready to hand in by doing them in solo. When they're ready to hand in get your 3 people in game and hand them all in and your achievement should pop.
Edit: (Thanks to Level 75 Noob)
"You can begin a new game from the main menu, beginning at key story quests. Which means that you can start a fresh game from the final mission! All of the first tier side missions are available from the get go. It'll make it a whole lot easier to stack up 5 quests ready to turn in"