I found a way which significantly reduces the amount of luck you need to get this achievement.
First, play with four controllers, and play an Original game, not Remix. Once the game's loaded, and the game instructions are displayed, you'll have the option to choose a Quick game or a Full Game. Choose Full Game.. Have the profile you want to get the acheivement signed into Controller #1. You can use ANY controller (Guitar, Drum, Microphone, Skateboard), as long as they have D-pad buttons on them.
At the start of the game, spin the spinner with all four players, if you're lucky your main profile will get a low number (1, 2, etc), and the other three controllers will get higher numbers. With all four controllers, you'll want to take the job route at the start of the game, instead of the educational path.
You'll want to try and get a minimum of two job options for each controller that is NOT getting the achievement (To get three job options, you must get ALL six green envelopes in the mini-game. To get two job options, you need 3, 4, or 5, green envelopes).
This is because no two players can have the same job. You need to have at least two job options for each player so that if a controller not getting the achievement accidentally chooses the police officer (obviously, all the jobs are hidden until the card is picked), that controller has an alternative to choose, so that the police officer is still free for the controller that's getting the achievement.
If you are lucky enough to get the lowest number right at the start of the game (i.e. you're the 4th player to spin the spinner), and you can get all six green envelopes in your mini-game, then you are GUARANTEED to pick the police officer. However, if you are 2nd person, that means one job of the six will be gone, leaving you only a 3/5 chance (60%) of randomly selecting the police officer, assuming you manage to pick all six green envelopes. If you're the 3rd person, that means two jobs of the six will be gone, leaving you a 3/4 chance (75%) of randomly selecting the police officer, assuming you manage to pick all six green envelopes.
Once the police officer is chosen, it's just a matter of playing the game until one of the other three players spins a 10. As soon as it happens, assuming your profile getting the achievement hasn't lost his job as a police officer from one of the job-change spots on the board, you'll get the achievement the moment the 10 appears on the spinner.
Please, if you decide this solution is not worthy of a positive vote, please say why before voting negatively, so that I can rectify the solution, and get a positive vote, thanks.