Let’s first compare three sentences without ところだ or ところだった.
It will become necessary for Kurumi to search for how to get rid of a corpse.
(Kurumi will have to search for how to get rid of a corpse in the future.)
It has (just) become necessary for Kurumi to search for how to get rid of a corpse.
(Kurumi now has to search for how to get rid of a corpse because of a recent change in the situation.)
It had (already) become necessary for Kurumi to search for how to get rid of a corpse.
(Kurumi was already in a situation where she had to search for how to get rid of a corpse because of an earlier change in the situation.)
They were a bit hard to translate because of なる. I resorted to beginning each sentence with “it” so the verb “become” would fit well to express a change in the situation at various points in time. The translations may look awkward as a result.
Anyway, these are precisely the situations Kurumi would have found herself in if someone had seen her, as described from a third person’s perspective at the time of the event. This should make sense because if ところ means anything, it is “situation” or “circumstance”, and the part before it describes what kind of situation or circumstance it is. (We had a similar discussion about つもり here.)
Conceptually speaking, adding ところだった at the end of each Japanese sentence would be like adding “The situation would have been such that …” at the beginning of its English translation and back-shifting the tenses inside the that-clause, whereas adding ところだ would correspond to beginning the English sentence with “The situation would be such that …” The former is used to look back at a counterfactual situation in the past, while the latter describes one in the present. (The past tense may also be used to describe a present situation, but let’s leave it aside here.)
The distinction between 〜るところだった and 〜たところだ seems to get blurred when the part before ところだ describes a state, rather than an action or a change of state, that would have resulted from said counterfactual condition, and that’s usually the case when 〜ていたところだ is used. The reason the distinction gets blurred could be because the same state would have continued till the present, making it unimportant whether the reference point is set at a past time or in the present.
The situation would have been such that Kurumi now had to search for how to get rid of a corpse.
(Kurumi would have had to begin to search for how to get rid of a corpse.)
The situation would be such that Kurumi was already in a situation where she had to search for how to get rid of a corpse.
(Kurumi would have already had to search for how to get rid of a corpse.)
Although we haven’t considered it so far, 調べねばならなくなっているところだった also works fine. Actually, this is the direct result of swapping the position of the tense marker on 調べねばならなくなっていたところだ.
While 調べねばならなくなったところだ may not be totally unacceptable, it doesn’t seem to work so well as the other options probably because 調べねばならなくなった is more naturally understood as describing a past change than its resulting state, and therefore, it doesn’t sound like referring to a counterfactual situation that would have lasted till the present as much as 調べねばならなくなっていた does. It sounds more like a normal expression of aspect.
I guess 空に飛ばしてやったところだ in one of your comments is OK because its focus is not so much on a past act of throwing as it is on its result. The person in question would be in the sky by now.
In general, I would say 〜ていたところだ sounds more natural than simple 〜たところだ when expressing a counterfactual situation. In the link you provided, every example that matches 〜たところだ also matches 〜ていたところだ, and they can be rephrased using 〜ているところだった without changing their meanings much.