Summary - Read the conclusion at the bottom for short answer, where various categories of Shastras - Upanishads, Grihya/ Dharma Sutras, Puranas, Itihasa; unanimously prescribe the wearing of the Yajnopavita and Shikha while following Sannyasa. When so many categories of shastras are saying the same thing, there can be no bigger Shastra pramana.
Firstly it is pertinent to note that as described here, Sri Vaishnavas are Tridandi Sannyasis, or those who carry three sticks. We will find many such references from various categories of scriptures.
1. The Minor Upanishads
Though some minor Upanishads may prescribe abandonment of the Yajnopavita and Shikha, from this answer, one can see that some minor Upanishads even prescribe the Yajnopavita and shikha for a sannyasi. Quoting the relevant portion:
Bahudakas are those who carry a triple staff, a water pot, a sling , a shoulder yoke, a water strainer, a bowl, shoes, and a seat; who wear a topknot, a sacrificial string, a loincloth, and an ochre garment; who beg their food from virtuous Brahmin households; and who thus seek the self.
One can also refer : this
2. The Sutra Granthas
Bodhayana sutra while talking of items to be renounced doesn’t mention renunciation of the Yajnopavita, instead it prescribes optional Shikha but also compulsory Sandhya vidhi possible only with the Yajnopavita (and Shikha):
अनग्निकस्तु मुण्डी शिखी वा अथ प्रक्षाळ्य जलार्द्रेण कर्पटेन सशिरस्कजलापकर्षणं कृत्वाऽन्तर्वासाः परितस्सन्ध्यामुपासीत॥१२॥
Having washed with a wet cloth and pulling out the water with the head cloth let him return indoors and properly perform the twilight ritual.
-Bodhayana Grihya Shesha Sutra, 4th Prasna, Chapter 16
The importance of Sandhya for a Sannyasi, using Veda mantras (and therefore having Yajnopavita) has also been mentioned by the Bodhayana Dharmasutra:
After performing his evening devotions by (reciting the verses called) Vāruṇīs, and his morning devotions by (reciting the verses called) Maitrīs.
-Praśna II, Adhyāya 10, Kaṇḍikā 18 sutra 21
[From Footnotes]: The Maitrīs occur Taitt. Saṃhitā III, 4, 11, 5, and the Vāruṇīs follow them immediately.
Agniveshya Grihya Sutra on the other hand is very explicit about not giving up either the Shikha or Yajnopavita:
अथातः संन्यासविधिं व्याख्यास्यामः .... यतेर्धर्मान् सम्यक् परिचरेत् नित्यशिखी नित्ययज्ञोपवीती नित्यसंध्योपासी नित्यप्रातःस्नायी नित्यं भैक्षाशी नित्यं यमनियमयन्त्रव्रतोपवासी....॥११॥
Now we are explain the Sannyasa ritual..... May he properly follow the Dharma of a Yati, [he is] always with the top-knot, always with the sacred thread, always a performer of the twilight ritual, always bathes in the morning, always one who eats by begging, always practises rules and restraint.
-Agniveshya Grihya Sutra, 2nd Prasna, Sutra 11
3. The Puranas
I’ve picked the Skanda Purana for a very special reason, as stated below. Quoting only the relevant portions from the Skanda Purana Suta Samhita Jnanayoga Khanda, Chapter 6:
कुटीचकश्च संन्यस्य ... शिखी यज्ञोपवीती स्यात्त्रिदण्डी सकमण्डलुः - of the Kuticaka Sannyasi..... may he be one with the top-knot, the sacred thread, three sticks and with a pot. (Verse 3-4)
बहूदकश्च संन्यस्य ... शिखी यज्ञोपवीती च देवताराधने रतः। - Of the Bahudaka Sannyasi... [He should he] one with the top-knot, the sacrificial thread and engrossed in worship of God. (Verse 6, 10a)
For the Hamsa nothing of either abandoning or leaving the Yajnopavita is mentioned but Sandhya is compulsory for them too (verse 14 and 27) and hence it’s safe to say abandonment of Yajnopavita is not prescribed. For Paramahamsa Sannyasis abandonment of Shikha, Yajnopavita, and Nityakarma is mentioned but at the same time, they’ve been ordained to recite Om (advised to chant after Diksha) during Sandhya (Nitya Karma time) so abandonment verse is probably a small error.
Although Sri-Vaishnavas wouldn’t be following the Skanda Purāṇa, all Puranas being from the same source, the rules of Sannyasa don’t change much and hence this can serve as an example of retention of the Yajnopavita and Shikha in the 4th ashrama. If a purana which is not followed by them, also gives the same procedures they follow, it shows the authenticity of their practice. (Though ultimately all Puranas have the same
4. The Srimad Ramayanam
This is one of the most respected Granthas in the Sri Vaishnava tradition. The characteristics of the fourth order given here are surely authoritative. Here though the Yajnopavita is not explicitly mentioned, but the presence of the Shikha is sufficient proof (like the above examples where except for Baudhayana’s optional Shikha, both Shikha and Yajnopavita have been prescribed. The Yajnopavita is not optional)
When Ravana comes to abduct Sitaji, he is described as a Sannyasi and the dress code is as follows:
तदासाद्य दशग्रीवः क्षिप्रमन्तरमास्थितः। अभिचक्राम वैदेहीं परिव्राजकरूपधृक्॥ ३-४६-२
श्लक्ष्ण काषाय संवीतः शिखी छत्री उपानही। वामे च अंसे अवसज्य अथ शुभे यष्टि कमण्डलू॥३-४६-३
Quickly taking hold of intervallic time then the ten-headed Ravana approached Vaidehi adopting an aspect of a sanyaasi, a meandering Brahman friar. [3-46-2]
That sanyaasi is now cloaked in smooth silky saffron, his head-hair is neatly tufted, sandals are wooden, umbrella is on right shoulder, and on his left shoulder a shaft of sanyaasi is shouldered, onto which his handy water vessel is hooked.-
-VR, Aranya Kanda, Chapter 46, verse 3
One can refer the linked answer for the commentary given about the presence of the Yajnopavita.
Though there are some minor Upanishads which prescribe removal of Yajnopavita and Shikha, we can see from the above that some other minor Upanishads, authoritative Sutra Granthas, Puranas and Itihasa all unanimously prescribe wearing the Yajnopavita and Shikha. Therefore there is conformity between various categories of shastras and this is Shastra Pramana enough to wear the Yajnopavita during the fourth ashrama.