I'm of the opposite small camp, which can't continue or write a story UNLESS I'm getting it the best that I can the first time around, but...
Without reading all the other replies, one suggestion is you can set out to write something lousy on purpose. Really sit down with the intention, "I'm going to write the crappiest story I can!" Then see if the inner censor will either fritz out in shock or sit there and harp, "You can make that part lousier! Ew! What's wrong with you? You can totally mess this up even more if you really tried!"

(Seriously, I've never tried this but I'm curious whether the inner censor that works so hard to get things "right" would work equally hard to get a lousy story "wrong.")
Or you can just tell yourself you're going to write a lousy story--not the lousiest, as that might be considered a challenge to the inner censor, just a lousy story--and see if it still acts up so much.
If it continues to do so, you can try bargaining with it. Tell it it can have its say AFTER you've written, say, a thousand words or five pages or some such. Whatever seems reasonable. Ask it to leave you in peace until you've written that, then it can speak up as much as it wants, and you will then respectfully listen to it. I realize on these message boards I'm always suggesting things like talking to the inner parts of yourselves like they're real people, but you'd be surprised, sometimes it really works. Especially since the more you try to fight an inner censor, the harder it'll tend to fight back. (I learned this from having OCD--the harder you fight the obsessions and compulsions, the harder they hit you. It's like a Chinese finger trap.) Sometimes compromise is the only way to go.