Understand SIG manual's instructions applies to a normal status with the bolt locked into the barrel extension. You've described a very different condition with a plastic round jammed inside the breech. Seems unlikely your bolt is actually locked in the barrel extension as anticipated by the manual's instruction.
Removing MCX barrel is an easy, loosen 2 retained torx fastener procedure. Your particular jam may have placed the barrel extension in a bind, making the barrel difficult to remove, but it's an easy <5 minute task to try.
It's only a suggestion, something I would do with no risk of damaging the rifle. If you're uncomfortable that's fine, I'll suggest calling SIG customer service before going to a gun smith. Am sure they'll get you fixed. I mention SIG because while MCX looks like an AR15, it's action components are very different. You don't want a smith's initial MCX experiance to be on your rifle.
Sadly, seeing a photo of your broken receiver on Sig Talk wouldn't be the first we've seen here..