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In my opinion, the phrase "I would like to register for the tournament" sounds better than "I want to register for the tournament." And I don't see any problem with it; it's a perfectly grammatical and adequate option.
By the way, regarding the topic of "Soviet-style" language: in English, the expression "Please, sign me up for the tournament" is usually used, which literally translates to "Please, register me for the tournament." And yes, this is a request. Moreover, it's a polite request (horrible, right?). And if we approach it very formally and meticulously, "want" is not a request to participate (although it may be implied), which is true both in Russian and in English.
In my opinion, the optimal option is: "Please, register me for the tournament."