As far as I know there isn’t anything in the earliest texts that would tell us exactly what sort of connotations bhagavā might have had for the disciples of the Buddha who used it, beyond the obvious fact that it was expressive of high reverence. The earliest indication is given in the Niddesa, a canonical commentary preserved in the Khuddaka Nikāya which gives a list of eighteen niruktis (explanations of a word’s meaning via ersatz etymologies). These are:
He has demolished lust, thus he is the Bhagavā.
He has demolished hatred, thus he is the Bhagavā.
He has demolished delusion, thus he is the Bhagavā.
He has demolished conceit, thus he is the Bhagavā.
He has demolished wrong view, thus he is the Bhagavā.
He has demolished the thorn, thus he is the Bhagavā.
He has demolished defilement, thus he is the Bhagavā.
Bhaji vibhaji pavibhaji dhammaratananti bhagavā.
He has divided up, analyzed and classified the Dhamma jewel, thus he is the Bhagavā.
Bhavānaṃ antakaroti bhagavā.
He has made an end to states of existence, thus he is the Bhagavā.
Bhāvitakāyo bhāvitasīlo bhāvitacitto bhāvitapaññoti bhagavā.
He is developed in body, virtue, mind and understanding, thus he is the Bhagavā.
Bhaji vā bhagavā araññavanapatthāni pantāni senāsanāni appasaddāni appanigghosāni vijanavātāni manussarāhaseyyakāni paṭisallānasāruppānīti bhagavā.
He resorts to remote lodgings in forests and woodlands, places with little sound, little noise, far from the crowd, isolated from people, suitable for retreat, thus he is the Bhagavā.
Bhāgī vā bhagavā cīvara-piṇḍapāta-senāsana-gilānapaccayabhesajja-parikkhārānanti bhagavā.
He partakes of robes, almsfood, lodgings, and medicinal requisites, thus he is the Bhagavā.
Bhāgī vā bhagavā attharasassa dhammarasassa vimuttirasassa adhisīlassa adhicittassa adhipaññāyāti bhagavā.
He partakes of the taste of the meaning, the taste of the Dhamma, the taste of liberation, the higher virtue, the higher mind, the higher understanding, thus he is the Bhagavā.
Bhāgī vā bhagavā catunnaṃ jhānānaṃ catunnaṃ appamaññānaṃ catunnaṃ arūpasamāpattīnanti bhagavā.
He partakes of the four jhānas, the four measureless states, the four formless meditative attainments, thus he is the Bhagavā.
Bhāgī vā bhagavā aṭṭhannaṃ vimokkhānaṃ aṭṭhannaṃ abhibhāyatanānaṃ navannaṃ anupubbavihāra-samāpattīnanti bhagavā.
He partakes of the eight emancipations, the eight bases for overcoming, and the nine attainments of sequential dwelling, thus he is the Bhagavā.
Bhāgī vā bhagavā dasannaṃ saññābhāvanānaṃ dasannaṃ kasiṇasamāpattīnaṃ ānāpānassatisamādhissa asubhasamāpattiyāti bhagavā.
He partakes of the ten developments of perception, the ten kasiṇa attainments, concentration through mindfulness of breathing, and the attainment of unattractiveness, thus he is the Bhagavā.
Bhāgī vā bhagavā catunnaṃ satipaṭṭhānānaṃ catunnaṃ sammappadhānānaṃ catunnaṃ iddhipādānaṃ pañcannaṃ indriyānaṃ pañcannaṃ balānaṃ sattannaṃ bojjhaṅgānaṃ ariyassa aṭṭhaṅgikassa maggassāti bhagavā.
He partakes of the four establishments of mindfulness, the four right kinds of striving, the four bases for spiritual potency, the five faculties, the five powers, the seven factors of enlightenment, and the noble eightfold path, thus he is the Bhagavā.
Bhāgī vā bhagavā dasannaṃ tathāgatabalānaṃ catunnaṃ vesārajjānaṃ catunnaṃ paṭisambhidānaṃ channaṃ abhiññānaṃ channaṃ buddhadhammānanti bhagavā.
He partakes of the ten Tathāgata powers, the four grounds of self-confidence, the four analytical knowledges, the six superknowledges, and the six buddha qualities, thus he is the Bhagavā.
Later Pali texts added a few more niruktis and Sanskrit Buddhist treatises more still.
As an aside, I know another English monk who studied Anuruddha’s Abhidhammatthasaṅgaha under a well-known Burmese sayadaw. The text opens with the usual: Namo tassa bhagavato arahato sammāsambuddhassa. The monk told me that his teacher spent the first 2-hour lesson just explaining the meaning of namo tassa, followed by four 2-hour lessons explaining the meaning of bhagavato, three 2-hour lessons on arahato … etc., etc. After that, two weeks were spent going over Anuruddha’s 2-line colophon. It was nearly a month before the monk actually began learning any Abhidhamma.
Horses for the courses.
Ideally the task is to select words in the target language that when encountered by a native speaker of that language will give rise to the same thought that was in the minds of the speakers of the source language when they spoke the words that you’re now translating. But what those thoughts are likely to have been will depend on the genre and age of the text you’re translating.
For example, if it’s an EBT then what most translators will go for is a rendering that expresses a similar degree of reverence to that which theists have for their God: “Lord”, “Blessed One”, “Auspicious One”, etc. It’s not likely that they would base their translation on any of the eighteen niruktis listed above, unless they had some grounds for believing that one or another of them was of great antiquity and may have been in the minds of the Buddha’s contemporaries.
On the other hand, if you’re translating, say, a medieval Sinhalese or Burmese Abhidhamma treatise, then you may well opt for a translation based on nirukti #8 in the Niddesa’s list, reasoning that this is the sense of bhagavā that a writer on Abhidhamma would have been likely to prefer or at least would have had uppermost in his mind at the time of writing his treatise. In that case you might go for a rendering like “the Great Analyst”.
Or if it’s a Vinaya treatise, then you might opt for a rendering based on #10 or #12. If it’s a meditation manual, then any from #14 to #17.