Any of the look-through-a-hole tests would seem to have to be set up *very* carefully or the setup will just bias which of the two identical images you find first.

The parallax of two eyes and objects at different distances produces a pair of images and apparently (I guess) some people's brains tend to process-out one of them (like not perceiving the blind spots, I guess), but some of us very clearly always perceive both and if one is more dominant than the other it's a very, very subtle thing. Something you could only really determine with many repetitions of a very carefully set up test to pick out some tiny bias toward left or right. In practice apparently no one has worked up such a careful test, and if it's not just obvious without needing a statistical analysis then it's just 'no dominant eye.'