Do Srivaishnavites have the equivalent of “ghataka shrutis” to resolve the birth vs guna caste question?
The Vedas themselves declare that caste is based on ancestry (gotra), and hence birth.
There are several Sruti texts that state Varna is based on birth. I've listed a few below:
Chhandogya Upanishad 5.10.7 - Those whose conduct has been good here will shortly get birth such as a Brahmana (brahmana yonim), a
kshatriya, or a vaishya. But those whose conduct has been evil will be
born in evil births shortly such as the birth of a dog (shva yonim), or
as a pig, or a chandala.
The word used is "yonim", and it means birth or womb, and is used to refer to birth as a dog (shva), and pig, and since it's also used along with the names of castes, it must be taken to mean that one is born into those castes just like animals, so it doesn't merely mean being born into a woman who self-identifies as a Brahmin. It means being born into the womb (yoni) of a Brahmin, and a person born into such a womb naturally acquires Brahminical qualities due to gene transfer.
Maitrayani Samhita 1.4.1 - We know not if we are brahmins or non-Brahmins. ...Therefore, when recounting our gotra-pravaras, say that the
devas are our ancestors.
This verse from the Maitrayani shakha of the Krishna Yajur Veda is referred to in Jaimini's Mimamsa Sutras, and commented on by Shabara swami. The sutra reads:
On account of the failings of women, (there can be no certainty
regarding one's caste); specially as the son belongs to the
Shabara's commentary for the sutra:
It is difficult to know
if one is really a Brahmana; - and this is what is figuratively spoken
of as 'we do not know'; and the difficulty in knowing it for certain
is due to the 'failings of women'.
This vedic verse shows that caste is determined by gotra pravaras (ancestral lineages), and hence caste is inherited and based on birth.
If caste was determined by guna or self-identification/self-claiming, like the followers of Vivekananda and reformist Vedantins believe, then this verse would not mention gotra pravaras at all since it wouldn't be relevant.
And another verse is from the Ashwamedha section of the Taittiriya Brahmana, which reads:
‘When a Sudra woman is the mistress of an Arya, she does not seek
wealth for prosperity’. Therefore they (the priests) do not bestow
royal consecration on the son of a Vaishya woman."
This shows that only kshatriyas by birth are allowed to be consecrated as kings, and not other castes.
The pre-Ramanuja Sri Vaishnava scholar Yamunacharya has stated:
Every man has some Vedic rites for which he is not qualified: the Brahmin is not qualified for the royal consecration, the Kshatriya for the ritual drinking of soma.
So it is proven that the Vedas talk about a birth-based caste system.
Then what about those scriptures that say caste is based on guna?
Here is how Vedanta Desikan resolves it in his Rahasya Traya Sara:
In such passages as this: - "He should be considered as a Shudra", the great sages called men of higher castes by the words applicable to lower castes and vice versa merely to indicate the degree of consideration or respect to be shown. - page 301