The ideal way for Melo to come to the Bulls is via sign/trade, in which he convinces the Knicks he is not coming back under any circumstances and is willing to take a starting salary of 18.338 million from the Bulls.
How did I arrive at 18.338? See below
As you can see, the salaries of Butler, Dunleavy, Snell, and Randolph are not included. I look at two of these players as desirable trade targets of other teams (Butler and Dunleavy) and two as less desirable (Snell and Randolph, more so Randolph).
Looking at the landscape of teams with multiple trade exceptions, the Bulls could send one of Butler or Dunleavy and Snell or Randolph to a couple of teams with 2 large enough trade exemptions. These two teams would be the Celtics and Nuggets. Otherwise, there are different ways to get teams involved with only one large enough exception, such as bribing them with future picks, getting multiple teams involved, etc.
Also, I believe there is a clause Randolph cannot be packaged in a trade. However, when using trade exceptions, I believe each move would be treated as a non-simultaneous transaction, because you are sending two players separately to a team with two separate large enough trade exceptions.
So, if the Bulls can line up a couple of deals for these four players, so they and Melo can truthfully say he is willing to walk away for 18.338 million, and the Knicks would get nothing, I'm guessing they would take Boozer and some combination of Dunleavy/Snell/Butler/Future Pick, etc. I would think that with the Bulls leverage they could avoid including Butler, but if it has to be done, so be it.
By executing this transaction a as sign and trade then the Bulls would have the MLE at their disposal and could bring over Mirotic this offseason.