Our Purvacaryas had not stated anywhere that Anjaneya is amsha of Rudra and this cannot be a reason that Swami Desikan did not write any specific sloka. Swami did not write any stotras based on his own volition but as per the ordain of each Perumal.
Swami Desikan had written slokas about Anjaneya in his Sankalpa Suryodayam. He had also mentioned that Anjaneya resided in Gandhamadana mountain.
Srivaishanavas can certainly worship Siriya Thiruvadi like Periya Thiruvadi. We also chant Anjaneya dhyana sloka as stated in Ramayana. It is important how Anjaneya is consecrated. If Anjaneya is consecrated based on Vaikanasa or Pancharathra agama he takes the position as a devotee of Rama, we can definitely worship and if he is consecrated as an amsha of Rudra then we cannot worship.
That is why our elders always worship Anjaneya at Perumal temple as he appears alongside Lord Rama and they do not support worshipping stand-alone Anjaneya because they do not know the consecration details.
We should be devoted to him and chant his slokas with reverence, nonetheless.