Patrick Schwarzenegger -- DUMB ICE ... Threw Cubes at Club DJ

Patrick Schwarzenegger used a FAKE ID to get into a Hollywood club this weekend ... and was only kicked out after he pelted the house DJ with ice cubes ... sources connected to the club tell TMZ.

We're told ... Arnold's kid -- who's only 19 years old -- used someone else's driver's license to get past the bouncers at The Sayers Club in L.A. on Saturday night ... and it worked.

But once he was inside, we're told Patrick was a total pain in the ass.

Sources tell us Patrick was rowdy and obnoxious ... and at one point, he began throwing ice cubes at the DJ. (His father did play Mr. Freeze, after all).

Once the ice began to fly, we're told club honchos recognized Patrick -- realized he was underage -- and kicked him out ASAP.

After he got the boot, TMZ shot footage of Patrick ranting about wanting to beat up the DJ.

Patrick wasn't arrested -- but using a fake ID is a misdemeanor in California, punishable by up to a year in jail and a $1,000 fine.

His father probably knew that already ... since, you know, he was the governor.