In Vaishnava parlance we say Garudazhwar; but in an upanysasm, Pazhaveri Lakshmi Narasimhacar (Balaj) swami mentioned we should not address as Garudazhwar but Periya Tiruvadi as per sampradayam. Are there any reasons behind this? Is it wrong to say Garudazhwar?

In the expositions of our preceptors, when the term Azhwar is mentioned, it either refers to Sri Chakratazhwar or Shankhathazhwar, the Panchajanyam.
While Hanuman is known as Tiruvadi and Garuda is known as Periya Tiruvadi based on the commentaries of our preceptors in their texts.
It is not an error to call Garudazhwar, but our preceptors state Garuda as Periya Tiruvadi. Based on this Sri U Ve Balaji swami must have mentioned so.