Here are the problems.
First of all, no matter what you may have read or been taught, identification is always and everywhere an ASSUMPTION. You cannot prove your IV is valid.
Second, no matter what you may have read or been taught, the family of Sargan-type tests are tests of OVER-IDENTIFICATION, not identification. You can "pass" the test and still not achieve valid identification.
Third, passing the tests, useless though they are, in any realistic fashion does not mean failing to reject the null at the .05 or even the .10 level.
The reason why is that our worry is we might fail to reject a false null. This is type II error. Choosing .05 essentially MAXIMIZES the chances of committing a type II error (minimizes the power of the test). I'd like to see p-values on the order of at least .25 to .30 (or higher).
Since identification is done by assumption, theory becomes super-important. The right way to do this in my view is by recognizing that the equation you seek to estimate is part of a system and the properties of that system will let you know whether identification is achievable or not.
If not, too bad. Estimate a reduced form and be happy.
I pretty much refuse to let my grad students go on the market with an IV in the job market paper. No way, no how. Even the 80 year old deadwoods in the back of the seminar room at your job talk know how to argue about the validity of your instruments. It's one of the easiest ways to lose control of your seminar.
We've had really good luck placing students who used Diff in diff (in diff), propensity score matching, synthetic control, and even regression discontinuity. All of these approaches have their own problems, but they are like little grains of sand compared to the boulder-sized issues in IV.