I can't tell, from your post, what kind of propositions you are trying to convince yourself of. If it's an attempt to win competitions, then you're putting your effort in the wrong place. Whether you win any given competition is largely going to be determined by who else shows up to compete. Improving your chances means reducing the number of people who can reliably beat you, and that only happens through research and practice (since murdering competitors is generally seen as bad sportsmanship).
Other than that, it sounds like you've discovered the flaw in Pascal's original wager (well, one of its flaws anyway). You can decide it's rational to believe something, but actually believing it is a different matter. In religion, actual belief is key, and therefore Pascal's wager isn't going to make a lot of true converts, even though it's a beautiful piece of reasoning.
I am having a similar issue, and am currently dealing with it by developing better acting skills. As long as I do and say things consistent with the belief set I wish I had (and don't), my ends are achieved regardless of whether I actually hold that belief set. This may or may not be applicable to your situation.
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Does the idea that it is a good thing to subject our beliefs (and even our belief in belief) to logical and analytical scrutiny count as belief in itself or is it so justifiable as to count as knowledge? If so, what is the justification?
I don't think it does. Scrutinizing your beliefs is a corollary - it naturally follows if you believe that "Truth is good and valuable and its pursuit is worthwhile." We value truth, we want our maps to match the territory, and so we scrutinize our beliefs. If anything needs to be justified, it's the value placed on truth and knowledge thereof.
And that's actually an interesting problem. Although my intuition shouts TRUTH IS GOOD, there's not much I can say to prove that statement, outside of "It's useful to be able to make accurate predictions." It seems like the goodness of truth ought to be provable some way. But maybe it's just a moral value that our (sub-)culture happens to hold particularly strongly? Perhaps someone better versed than I am in the arts of rationality can give a better answer.