Unique SAT complexity clarification
To prove coNP-hardness, you don't need a certificate (that would be needed only if you wanted to show coNP-completeness). We just need some coNP-hard problem that reduces to UniqueSAT. UniqueSAT is indeed likely not in coNP.
You can ask an oracle multiple queries. You could begin by asking if the formula has a satisfying assignment (and, using self-reduction, find a solution if one exists). Perhaps you can then use further queries to determine if another solution exists?