Using approximations to optimization problems for threshold problems
First off, a technicality: optimization problems aren't NP-complete; only decision problems can be NP-complete (or in NP).  Thus, it's only the decision problem (what you call the "threshold problem") that is in NP, or NP-complete.

On to the substance of your question.  The answer is: You can't.  In general, the approximation algorithm typically doesn't let you solve the decision problem.  Think about it.  We have many problems for where the decision problem is NP-complete, but where we have a polytime approximation algorithm.  If the approximation algorithm let us quickly solve the decision problem (or the corresponding optimization problem), then we'd immediately have a polytime solution to a NP-complete (or NP-hard) problem, and thus a proof that P=NP.  When you start with some assumption and it leads to a conclusion that is too good to be true, that's a sign that the assumption probably wasn't valid.

No, your "obvious answer" doesn't work.  Decision problems aren't allowed to return "I don't know".  No, there isn't a better way (assuming the problem is NP-hard, and assuming $P\ne NP$).