Reading your second last paragraph, you seem to say that the propositions ’Absence of evidence is evidence of absence’ and ‘Absence of evidence is not evidence of absence’ are not opposed to each other because ‘they are not saying the same things.’
If we plotted them on the Square of Opposition, a lot would depend on whether we took them to be universal propositions or particular propositions. If we took them to be universal, then they would be contraries, and contraries cannot both be true. However, if we took them to be particular, then they would be subcontraries, and subcontraries cannot both be false but may both be true.
Perhaps the addition of a temporal adverb might clarify matters. So
‘Absence of evidence is evidence of absence’ would become
‘Absence of evidence is [sometimes] evidence of absence
as in the case of my elephant in the room example.
‘Absence of evidence is not evidence of absence’ would become
‘Absence of evidence is not [always] evidence of absence
as in the case of my subatomic particle example.
You write: ‘If I am understanding this Bayesian thing correctly, it is saying the longer something goes without evidence the less likely it will be true. I don’t see how one could get there from absence of evidence is evidence of absence.’
Once again, I believe material [as distinct from formal] factors come into play here. Everything depends on the would-be obviousness of what it is that one seeks evidence of in relation to the relative finitude of the search area.