Classic good-cop-bad-cop is definitely wrong with a toddler because it involves a lot of lying. The bad cop threatens to beat the suspect up or otherwise do something cops are not allowed to do. The bad cop steps out and the good cop says "I'm on your side dude, but that guy is out of control and I don't know what he'll do next. Listen, if you just [....] I think I can get you out of this questioning session and maybe save your skin." Everything about this is a lie. The good cop isn't on the suspect's side, isn't trying to help the suspect, and knows full well the bad cop isn't ever going to hit, torture, or whatever the threat is. None of this is appropriate with your child.
So let's take a milder version of it. Toddler has made a mess. Last time this mess was made, Daddy yelled a little. Should Mommy say "hurry, let's tidy this up before Daddy comes home and sees it?" No way.
OK forgot trying to change their behaviors. How about Mommy puts toddler in timeout and afterwards, Daddy lets toddler out with lots of cuddles and fun things to do? Or Daddy puts toddler in timeout and Mommy is the cuddles and fun one? This really isn't any better, is it?
What I would urge you to do is to give away any thoughts of an adversarial stance. You want your child to learn to be in control and not to tantrum. You want your child to comply with reasonable requests, whether that's putting on an item of clothing, helping to clean up, not hitting, or anything else. When your child can do those things, not only will you be happier but your child will too. You don't need to break their spirit, make them give up their resistance, force them to do what you want. You need to show them how to do the things everyone wants them to learn to do. Nobody is a bad cop in that. Nobody is a good cop. You can love your child while you're putting something out of their reach. You can smile and cuddle someone as part of letting them know they are not allowed to do a particular activity. You can be happy and cheerful while carrying a shoeless coatless child to the car (with their shoes and coat in your hands for later) and explaining there isn't time to play on the way now.
Whatever these rules are, they should be the rules FOR everybody and BY everybody. If the toddler can't eat before dinner, nor can Daddy. If feet aren't allowed on the couch then that's the rule, not just "if Mommy's looking." This kind of consistency isn't cruel or ganging up. You are showing your child the "laws of physics" of your home. This is how things go. Trying to understand mommy rules, daddy rules, sitter rules, grandparent rules - that's complicated! Remember, abiding by these rules isn't something mean or unfair, it's how life works in your house. And the toddler is trying to learn that. Make it as smooth as you can, and that includes being consistent, and not seeing the rules as mean.