Though it has been reported that LeBron is seeking a full max salary this summer, the guess here is he's willing to take slightly less than that because of all of the money he earns from endorsements (he earned a reported $53 million last season alone). Even if he cared about maximizing income above all else (which we doubt), leaving a few million on the table to show his "loyalty" to Miami would more than pay for itself through enhanced goodwill with the public (and potential endorsees) and help give the Heat more room to fill out his supporting cast.
The following scenario outlines one such path to roster upgrade. It has the team (a) signing free agent Kyle Lowry as a massive upgrade at point guard; (b) bringing back Chris Andersen and Ray Allen on reasonable short-term deals; (c) retaining Rashard Lewis and Udonis Haslem on minimum contracts; (d) recruiting massively under-rated veterans like Ekpe Udoh, Vince Carter, and Anthony Tolliver to play for the veterans' minimum; and (e) using the bi-annual exception to snag C.J. Miles as a quality backup wing.
This isn't the only combination that works for the Heat to improve, but it's one that sticks closely to news reports and the Miami M.O.