The problem is not the carburetor (or fuel injection for that matter). If it was, the engine would cut out much more quickly. Instead the problem is with the fuel feed from the tank, if you fly inverted for too long, the feed lines aren't submerged in fuel and the lines will run dry. The amount they hold will give you some leeway, but you can't simply fly inverted for prolonged times.
Even if the fuel system could theoretically handle this, you'd still be faced with the problem that your oil pump would run dry after a short time, which would cause a drop in oil pressure, wreaking havoc on the engine.