For a sample of contemporary usage, I searched Google News.
"Indeed, it gives insurers almost no other choice but to stop covering abortion..." --Vox, US
"Jenkins felt she had no other choice but to resign," --Minneapolis Star-Tribune
"'That was an offering to people who didn't have an opportunity to have a livelihood - who had no other choice but to beg,' Baatz said." --Colorado Springs Gazette
"based on what we saw, we have no other choice but to cast this early proclimation:[sic] Wonder Woman will indeed be the kick-off to a brand new era of DC Films projects."--comicbook.com
"With limited cap space, a ton of needs and 24 unrestricted free agents on the dock, the Bills may have no other choice but to let their No. 1 cornerback pursue greener (literally) pastures in free agency."--Website about the Buffalo Bills football team
"Since Doctor Strange had no other choice than to bargain with Dormammu..."--moviepilot.com, original Berlin-based, it seems.
"The sheriff said the deputies 'had no other choice but to shoot the man.'"--Greenville News
"I have no other choice than to assert my right to live,"--The Guardian (UK)
"We are so good when our backs are against the wall and we are so good when we have no other choice but to win..."--The West Australian
"'I thought there was no other choice but Rachel and she's a Dallas girl, so we're going to keep that trend rolling,' JoJo exclusively told E! News"
My tentative conclusion is that in the US, "not other choice but..." is overwhelming more common than anything else.