Did he ever demand that Polk be removed? He didn't get on with Polk, but rather than force the issue he left Polk in command of a third of his army.
He definitely tried after Chickamauga, I believe after Stones' River.
Davis didn't do diddly squat to support Bragg in this regard, however, so what was Bragg to do?
Kinda hard for any Confederate general to remove officers when Davis is standing by the officers in question.
Edit:
On March 30, 1863, Polk continued his behind the scenes manipulation to have Bragg relieved of command. Polk wrote to Davis recommending that Bragg would be of greater value in some other field. He recommended Bragg for Inspector General. On March 30, 1863, Davis decided to visit the army himself and found the army in a high state of efficiency. In the mean time, the army had also gained 9,414 men. According to Parks, Bragg blamed Polk for the disaster at Perryville. On April 13, Bragg sent to all wing and division commanders, except Polk, a letter citing Polk’s note from Bardstown and a section of his report of the Battle of Perryville. Bragg wanted to prove Polk’s disobedience to his officers. Hardee sent his letter to Polk, fearing that Bragg was planning a court martial.
49 According to historian Nathaniel Hughes, Bragg accused Hardee of “sustaining Polk in his disobedience” at Bardstown and Perryville. Hardee refused to answer.
50 Cozzens says that Davis merely deferred to Johnston’s judgment and by mid April 1863, the matter was closed.
http://www.bryansbush.com/hub.php?page=articles&layer=a0409
Bragg being Bragg, I doubt proving Polk's disobedience would have been merely a point in an abstract game.
So he should have sent a few more dispatches?
And y'know, obeyed orders. This is not something to trivialize.
It was a great barrier to the AoT's chances of success that the senior subordinate to Bragg acted as - to put it charitably - an independent commander.
Were his troops broken and driven off the field, like Van Dorn at Pea Ridge?
Did he march a division into an ambush, like Sumner did at Antietam?
Did he ignore threats to his flank, like Burnside at Antietam, or Howard at Chancellorsville?
Did he go off and leave his men without command, like Pickett at Five Forks?
I'm not saying Polk was a particularly good commander. But I don't see where he did particularly badly, even once, much less throughout his service.
Apparently "being just short of open mutiny" doesn't count, but he behaved short of "no worse than average" in those battles - and there are no examples of him doing particularly well as a counter (unlike Burnside or Howard, who have a few successes to their name).
Not having his men routed is not the same as being able to deploy them effectively or organize assaults effectively or even proficiently.
So I strongly suggest reading Thomas Connelly's books on the Army of Tennessee to see Polk's consistently shitty behavior - even if he was a tactical genius his malicious insubordination should have seen him shot.
And throwing away good opportunities to attack by being insubordinate is arguably as bad for the Confederacy as throwing them away by any other form of unfitness.