Tactical voting when the voting rule is unknown
You use the term 'deterministic' several times to refer to Gibbard-Satterhwaite, and a related question might specify preferences over risk for voters, and then ask whether there's a non-deterministic voting rule for which truth-telling is incentive compatible. I think this paper, by Gibbard from 1977, is relevant to that question.

As for the question you actually asked, if I understand you correctly, 'truth-biasedness' seems pretty strong. A voter misreports only if both (a) there are no voting rules under which misreporting would hurt her and (b) there is at least one voting rule under which misreporting would help her. So what's wrong with the following?

Say you have 3 voters, and consider 3 voting rules, one for each voter. Voting rule $i$ is: "Choose voter $i$'s most preferred alternative, unless the other two voters both prefer the same alternative and it is not the same as voter $i$'s most preferred alternative." 

If the other two voters report truthfully, then a truth-biased voter can't gain from deviating: If voting rule $i$ is in effect and the other two voters differ in their most preferred alternative, then she cannot gain from misreporting because she gets her favorite alternative. If the other two voters both prefer the same option, then regardless of the voting rule that option will be chosen. In the former case there is one voting rule under which misreporting hurts her, in the latter case there are no voting rules under which misreporting helps her.