Barkley is a Republican.
EDIT: I think you're right that he's an independent but he seems to lean slightly right.
Either way, there is no universe in which he's a Democrat.
I don't think he's supported the GOP since Iraq though.
Yes he has. In 2016, he supported Kasich for President (he subsequently went #NeverTrump after Kasich dropped out).
That’s the only time I know of, at least since 2004, that he’s endorsed a candidate at all.
“The Republicans are full of it. The Democrats are a little less full of it." - Charles Barkley, circa 2007
“I'm serious. I've got to get people to realize that the government is full of it. Republicans and Democrats want to argue over stuff that's not important, like gay marriage or the war in Iraq or illegal immigration... When I run — if I run — we're going to talk about real issues like improving our schools, cleaning up our neighborhoods of drugs and crime and making Alabama a better place for all people.” - Charles Barkley, circa 2006