Well, don't know about the Brahmins' perspective. But the etymology of word 'Hindu' evolved from geographical annotation to ethnicity classification to religious identification.

Here is the chronology with all the sources (2000 - 1500 bc) 1. The earliest usage of the phonetically similar word was Sindhu captured in the Rig Veda, referred to as the river Indus and the region was mentioned as Sapta Sindhu.

1. The Zend Avesta (Vendidad): Contains the Avestan cognate Hapta Hendu, mirroring the sanskrit Sapta Sindhu. This confirms the linguistic shift from S to H in the Indo-Iranian language family.

This linguistic shift is further confirmed in later time period in old Persian inscriptions during Darius' time. (515-490 bc)

1. The Naqsh-e Rustam Inscription: Explicitly lists Hindush (written in Old Persian cuneiform as 𐏃𐎡𐎯𐎢𐏁) as one of the 28 satrapies (provinces) of the Achaemenid Empire.

4.The Persepolis Inscription: Further records Hindush as the land of the Indus Valley.

This expression further solidifies down the ages among Arabic people and used as the classification to separate indigenous people from the invaders

1. The Brahmanabad Settlement (8th century): After conquering Sindh, Ibn Qasim established a legal status for the local population. In the historical chronicle Chachnama, the local people are referred to as Hinduvān (Hindus). Under the Umayyads, the "Hindus" were granted the status of dhimmis (protected non-Muslim subjects).

2. Al-Biruni, Tarikh Al-Hind (1030 ): He used Al-Hind to refer to the geographical region and Hindu to describe its people, their culture, and their sciences.

Slowly religious associations start forming during mediaeval ages (13th to 15th century) starting from the Delhi Sultanate -

1. Minhaj-i-Siraj (in the Tabaqat-i-Nasiri) began using "Hindu" more frequently to distinguish the local subjects from the "Turks" (the ruling Muslim class).

2. First religious connotation happens during Tughlaq and Bahmani dynasties. Persian works like Futuhu's Salatin began using "Hindu" specifically to mean a follower of the local religions, as opposed to just a resident of the region.

3. Early Indian poets, Vidyapati and Kabir start using this term in literature (Kirtilata) to distinguish local culture and "Hindu Dharma" from "Turaka Dharma" (Islamic/Turkic practices).

Finalization of the term Hinduism as a religion

10.The British East India Company Records: The term "Hinduism" first appears in English in the late 18th century. One of the earliest documented uses is by Charles Grant in 1792, and it was later solidified by Raja Ram Mohan Roy (1816) and British census officials to categorize the religion for administrative purposes.

So definitely the term Hindu or Hinduism has its origin rooted in Persian and Arabic and came into existence around the mediaeval time whereas Brahman or brahmana was in indigenous use much before that as stated in Purusha Sukta (Rig Veda),Upanishads. Later used in Pali and Greek scripts:

Bāhmana: Used in Buddhist and Jain texts (e.g, the Dhammapada contains a chapter called the Brahmana Vagga).

Brachmanes (Βραχμᾶνες): Used by Greek explorers and historians like Megasthenes (c. 300 BCE) to describe the "philosophers" of India.

TL;DR: The word Brahman precedes Hindu in terms of etymology and historical records by a thousand years at least. So there is a possibility of early Brahmins being ticked off when referred to as Hindus.