This would be a case of "desperate times call for stupid measures".
Barkley was a great player, he's entertaining on television, but I have no doubt he'd be the first player to trade away a star player because he wrote the wrong name when submitting a trade to the league office.
He's a clown and he is almost always wildly incorrect in his prognostications.
I love the segment they do on Inside the NBA where they quiz Barkley on if a player is still in the NBA and what team he plays for. He basically strikes out every time it's done. Barkley may know how to play basketball himself but he would be the last TV personality I would hire to run a team. As bad as it may be I would prefer Stephen A Smith or Skip Bayless to Barkley. At least they try to stay current on league happenings.
Putting Barkley in charge of our change of identity would announce to the NBA and all of its fans that the Phoenix Suns days of being a passive, gimmick team are over.
Barkley himself would be the ultimate gimmick. No team without a history with him would consider hiring him for any sort of role outside of color commentator. There is nothing he's displayed on his time on Inside the NBA that would indicate he would be a quality front office executive.
I'm not even sure he would tell Sarver off, which is the true reason you want him as GM. You want someone who will tell Sarver he sucks and to stay away, regardless of other qualifications. The problem there is you have zero clue what happens behind closed doors. Barkley may mock Sarver when he's on TV now but if he were his employee I suspect that would change. One thing Barkley loves more than talking is money and Sarver would be the guy signing the checks. I have no doubt that if he paid Barkley enough the Chuckster would gladly endorse Sarver on national TV.