Mr Bishop will be arraigned on a charge of second-degree assault.
The Denver District Attorney's Office said Bishop could face additional charges pending the results of a blood alcohol content analysis.
Chase Bishop is pictured in this undated photo released by the Denver Sheriff's Department. He scrambles for the gun, which fires as he picks it up.
But as Bishop performs a backflip, a handgun falls from his waistband holster onto the dance floor.
The Denver Police Department responded to Mile High Spirits Distillery at 2201 Lawrence St. on the morning of June 2.
The FBI declined to comment on the case "in order to preserve the integrity of the ongoing investigation".
Video from the June 3 incident shows Bishop landing a backflip at the Mile High Spirits bar, and reaching down to pick up the weapon that had fallen from his trousers.
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He turned himself in on Tuesday morning after an arrest warrant was issued by Denver police, said Maro Casparian, a spokeswoman for the Denver District Attorney's Office.
The customer hit by the bullet was taken to hospital with a non-life-threatening injury.
Bishop is not assigned to the Denver office but works out of Washington, sources told KUSA.
The man who was shot was identified as Thomas Reddington and he said in an interview that he was able to laugh at the situation.
"I heard a loud bang and I thought some idiot set off a firecracker", Reddington told ABC News. "Then it clicked that I've been shot".
"It is shocking that the only shooting to ever occur at our establishment came about as a result of an Federal Bureau of Investigation agent entering our distillery tasting room carrying a loaded firearm without our knowledge, in violation of our rules", it wrote in a Facebook post.
The bar also said the patron who was shot is welcome to complimentary drinks forever at the bar.
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