Cleveland can only sign him to a four-year deal because they only have Early Bird rights.  It can be for the max with max (7.5%) raises, however.

I doubt owners will push back enough that it will affect LeBron.  If they want to do away with super teams, they should get rid of the max salary, which helps LeBron make more.  They could try to institute a hard cap, like the NFL, but that would create a huge fight with players at a time when the NBA is enjoying its highest popularity in history.  Furthermore, undoubtedly there are some owners who would be against a hard cap, because it would make it impossible for them to outspend their opponents, something they are willing to do.  The players would be unified against it, and owners would not be unified for it.

Accordingly, even if the cap is lowered (also unlikely in my opinion) LeBron could still sign for the max next season, because Cleveland would have Bird rights and be able to go over the cap to sign him.  I think he'll sign his five-year deal next year.  Right now, he's enjoying making Dan Gilbert grovel every summer.