This one is simple.
If the Bulls offer LaVine the 5 year max (and the only reason to think they won't is if their medical staff advises otherwise), he'll re-sign. Anything less than that and he'll minimally utilize the Spurs as leverage, if not sign with them altogether.
In addition to his friendship with Murray, they can sell him on being "the man" again, instead of a floor spacer for an aging, one dimensional player. They can also sell youth that can insulate him defensively, financial flexibility and extra picks as assets to improve the team going forward.
Any notion that the Spurs shouldn't sign him because of a sub championship ceiling is absurd. Almost every path leads to that, they've actively avoided giving themselves the best odds to create something greater and this is among the last places that can afford to turn down a prime, All-Star free agent, who's a good on court and likely off court fit too.