In his Seśvaramīmāṃsā ad 1.1.12, Veṅkaṭanātha explains that the example of proper names does not prove that language in general depends on convention. He writes that the case of proper names is not a dahanadṛṣṭānta, possibly ‘an example which sets on fire [the whole theory]’.

Why so? Because

प्रकृते यथोपलम्भं स्वभावसहकार्यादिनियमात्*

Because in the case at hand (i.e., language, composed of proper names and common words) there is a restrictive rule regarding the role of intrinsic signification and its auxiliaries according to how [linguistic expressions] are grasped.

In other words, convention, as an auxiliary of intrinsic signification, plays a role, but only in the case of proper names. And the distinction is possible based on one’s experience of linguistic expressions, insofar as linguistic expressions which are not proper names are immediately grasped as significant by people who know the language. By contrast, they might have doubts when they encounter proper names of unknown places or people.

These are, as usual, my thoughts only. Should you have a different understanding, you are welcome to share it. I would be happy and grateful to read it.

*(The editions add a च after प्रकृते, which makes the connection with the previous sentence unclear. The manuscripts provide a better reading, without the च).