So to get to the full max, given the scenario you outlined, we would indeed have to move Mills since I doubt we'll be willing to part with our younger players. On the bright side, that should give us enough room to offer LeBron the full max and use the space left over to bring in Milutinov and other free agents. Then fill out the rest of the roster with minimum players.
So our roster could be like this:
PG: Murray, White, Forbes
SG: Leonard, Ginobili, FA filler
SF: James, 18th pick / FA filler, FA filler
PF: Aldridge, Lauvergne, FA filler
C: Milutinov, 18th pick / FA filler, FA filler