For what reason did Dharmakirti argue that absences are conceptual constructions? For what reason did Dharmakirti argue that absences are conceptual constructions? I wondered if it was because real absences would have svabhava, would be essences, because they do not change in time?

I can't find anything to cite now, but I believe he argued this. I mean everyday absences, such as a cow from a field, rather than anything technical about apoha, which I am not that familiar with.
Say you walk into a room and look around and think to yourself, "Ah, this room has no elephant in it!"

The absence of an elephant in that very room is an object of knowledge. It can be known by a valid mind that there is - in fact - no elephant in that room. Since the absence is an object of knowledge it can be said to exist. So how does it exist? In order to answer this, back up and look at how it was known.

The absence of an elephant in that room was known merely by conception and inference on the valid basis of failing to perceive any elephant in that room and depending upon the thought which knew it. It is not a real, essential or inherent absence. How can we know this with certainty? Because if the absence of an elephant were real, essential or inherent, then everyone walking into that room would know that absence and think to themselves, "Ah, this room has no elephant in it!" which plainly does not happen. Moreover, it would be impossible for someone to - at any of the three times - walk an elephant into that room. If someone were to walk an elephant into that room, then this would annihilate the absence. But inherent things cannot be annihilated otherwise they would depend upon something else. Moreover, the absence of that elephant - were it inherent - would not depend upon anything at all.

No, the absence of an elephant in that room exists conceptually, dependently, relatively, in an illusory-like way without even a shred of inherent existence at all. In the same way, were someone to walk an elephant into that room, then that elephant would exist conceptually, dependently, relatively, in an illusory-like way without even a shred of inherent existence at all as would the "absence of the absence of an elephant."

This refutes real Abhava along the lines of what Dharmakirti might employ and is in agreement with Madhyamaka's like Nagarjuna. However, Dharmakirti was not a self-identified Madhyamaka per se and had some apparent disagreement with other points of doctrine, but on this point I think they agree.

If you are talking about Apoha, well that is related but different. That is getting into the epistemology of how nominal things are known from real and inherent - but unconceivable - entities. The upshot is that Madhayamaka's like Nagarjuna had apparent disagreement with Dharmakirti on this point in so far as Madhyamaka's like Nagarjuna did not think that anything at all was real or inherent even if inconceivable.

Hope this helps!