Is this that-clause a complement?

In that discomfort, breathing quicklime and tar, no one could see very well how from the bowels of the earth there was rising not only the largest house in the town, but the most hospitable and cool house that had ever existed in the region of the swamp.
(One Hundred Years of Solitude, tr. by Gregory Rabassa)

The that-clause is a complement providing the criterion for ‘the largest’ and ‘the most’, isn’t it? (It seems like the ‘indirect complement’ that is said in ‘The Cambridge Grammar of the English Language.')

No, the that-clause has no grammatical relationship with the largest. It's quite possible there was previously a larger house in the town (that's just been demolished, perhaps).

What the final clause modifies is specifically the word most, but it's complicated by the fact that strictly speaking there's a deleted second instance (the most hospitable and [the most] cool house). They're both being modified, even though one isn't actually there any more.

It's perfectly possible to describe something with just a superlative. For example,...

She was the sweetest girl.

...which can be modified it with a "that-clause"...

She was the sweetest girl that I had ever known.

That's all that's happening in OP's example. Without the qualification, the default meaning would be the most hospitable [and coolest] house ever, anywhere.