Team options have to be picked up or declined before free agency begins I believe. They can go over the cap to pay Livingston, but he's going to count something like $8.3 million against the cap because a team's free agents always count at least 150% of their previous year's salary against the cap until they are either signed or renounced. It's one of the loopholes the Spurs used with Danny Green and Kawhi Leonard last year to sign Aldridge. Green was only making $4 million so his cap hit was $6 million, and Leonard was making $3 million on a rookie deal and his cap hit was either $6 million or $7.25 million (I can't remember if the rule was 2x or 2.5x previous year rookie salary). So Green and Leonard signed last with Bird Rights.
What I see as the most likely scenario is they pick up their option on Livingston for another year (he's a bargain at $5.8 million), keep both Iggy and Bogut, decide to sign Barnes using Bird Rights (though he'd officially sign last) and then use about $10.5 million capspace they have (thanks to the Barnes $8 million cap hold) to go out and sign another bigman to replace Speights and Ezeli.