The structure would be that submitters are intended to submit proposals which are structurally like those of normal cipher competitions, e.g. make the same security arguments, etc. To focus efforts most usefully, they could be required to not just lie about the properties they claim (not a restriction on real attackers, but we already know that if people make claims that people accept and no one checks that this can allow backdoors-- and due to the risk of being caught real attackers would probably avoid outright lies).
But every entry is required to contain an exploitable algorithmic backdoor, not just a certificational weakness.
The process continues like a normal cipher competition, and the winner is selected based on no one being able to break their scheme.
Then the backdoors are disclosed. (Perhaps some escrow process could be used to make sure that they're really back doored, or a ZKP to that proves they could break a nothing-up-my-sleeve instance could be required along with the submission)
I think from such a process we could learn a lot about what submission norms are actually useful for protecting against cleverly constructed back doors, and something about the submission norms that we're missing.
Alternatively, it might be useful for normal competitions to allow secretly backdoored contributions. E.g. every submitter submits a commitment at the beginning, and then at the end everyone reveals their commitment (to avoid the case of a bad submission being revealed as "white hat" selectively only if its caught). ... but I fear that the result of such efforts would do more to erode our trust in the process (because it would fail to find the backdoors) than it would educate us about making things more secure. It would, however, make cryptanalysis of proposals more attractive, because it would increase confidence that there are discoverable weaknesses in at least some of the proposals.