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DeMarcus Cousins allegedly told his ex-girlfriend he'd put a "bullet in [her] f*cking head" if she didn't let their child come to his wedding ... and the woman claims she has the audio to prove it.

TMZ Sports has obtained court docs and a police report filed by DC's ex-girlfriend, Christy West -- claiming the NBA star threatened her on August 23, just one day before Cousins was set to marry his new girlfriend, Morgan Lang, in Atlanta.

In fact, West recorded the alleged threat and claims she played the audio for court officials ... and now we have a copy of that same recording.

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West claims she made the recording while at a trampoline park in Alabama with their 7-year-old son ... and she's adamant the voice on the other line is Cousins'.

"I'm gonna ask you this one more time before I take it to another level," the man says ... "Can I have my son here, please?"

Christy responds, "No" -- and that's when the man makes the alleged threat -- "I'm gonna make sure I put a bullet in your f**king head."

That's when the audio cuts off -- but Christy has since filed court docs in Alabama seeking a restraining order against Cousins claiming he ALSO told her that he was going to kill her, "even if he didn't have to get his hands dirty doing it."

In the court docs, Christy also claims Cousins had choked her during a previous incident -- but she didn't provide any specific details.

She wants officials to keep Cousins away from her and their son -- but she notes she's open to allowing Boogie to have supervised visitation.

Christy also went to the the Mobile Police Dept. and filed a report with cops. A police spokesperson tells us they have opened an investigation into the allegations.

As for Boogie's wedding, it went down on Saturday as scheduled in Atlanta -- and while a ton of famous people showed up (from Matt Barnes to Anthony Davis to Draymond Green), his 7-year-old was not in attendance.

We've reached out to Cousins for comment -- so far, no word back.