These are comments that are distilled from a review article by Edwin G. Pulleybank: "The Roman Empire as Known to Han China,"
Journal of the American Oriental Society, 119 (1999), pp. 71-79. The book he is reviewing is D. D. Leslie and K. H. J. Gardiner,
The Roman Empire in Chinese Sources.
First of all Chinese references to Da Qin only begin in the Later Han.
Pulleybank, citing the French sinologist Paul Pelliot, finds it almost certain that Da Qin is a reference to the eastern part of the Roman empire and not to Rome as a capital of the Roman Empire or to the Roman Empire as a whole. He also does not think that Da Qin is a phonetic transcription of a foreign word (in any language).
It's a rich article and I invite people to read it themselves. But here are some highlighted texts and discussion:
One main text for resolving this question seems to be the Sanskrit text of the "Questions of Milinda" (the Greco-Indian ruler Menander, c. 155-30 BC), or Milindapanha. This text was translated into Chinese during the Eastern Jin as Naxian biqiu jing (now in the Taisho Tripitaka [the Buddhis canon] 1670a and

.
Menander was conversing with the sage Nagasena. Nagasena asked the king:
"In what country was Your Majesty originally born?" The king said, "I was originally born in the country of Da Qin. The name of the country is Alisan."
Alisan is quite definitely Alexandria by Egypt (Pelliot argued against ID with other Alexandrias in the east), which makes perfect sense it seems. This is a piece of evidence he uses to support the argument that Da Qin references to the eastern regions of the RE.
Pulleybank then goes on to question the ability to derive Da Qin as though it came from a phonetic transcription from a non- Chinese word. He cites the first datable reference as the Hou Hanshu where the mission of Gan Ying in 97 AD is descrived.
"In the sixth year (of Yongyuan, 94 AD), Ban Caho again attacked and overthrew Yanqi (Karashar) and thereupon over fifty countries all offered hostages and submitted. Of them, Tiaozhi, Anxi, and the various countries reaching to the edge of the sea over 40,000
li distant, all offered tribute through multiple interpreters. In the ninth year ( 97 AD) Ban Chao sent his aide Gan Ying who got as far as to look upon the Western Sea and return. These were all places that had not been reached in previous ages and are not described in the
Classic of Mountains [and Seas] [Shan hai Jing], He gave a full account of their land and customs, telling of their precious and strange products. Thereupon distant countries, Menqi and Doule, came in submission and sent envoys with tribute." (
Hou Hanshu preamble to chapter on Western regions)
[it goes on in the same chapter]
"The Protector General Ban Chao and Chao sent Gan Ying on a mission ot Da Qin. He reached Tiaozhi, looked upon the Great Sea and wished to cross, but the mariners on the western edge of Anxi said to Ying: "The Sea is very broad and vast. With favorable winds those who come and go on it can cross in three months but if they encounter delaying winds it sometimes takes two years. Therefore those who set out on the sea always takes supplies for three years. Voyaging on the sea makes people long for sight of land and suffer from homesickness, and many perish." When Ying heard this he gave up his plan."
So Gan Ying never got to the territories of the Romans. The "Parthians" allegedly talked him out of it by stressing the length and difficulty of the journey.
The only plausible place for this conversation by the Great Sea/Western Sea to have taken place, if we accept the report, iis somewhere on the Indian Ocean (more likely) or on the western edge of the Persian Gulf.
Overall Pulleybank concludes in this way (p. 78): "A point that needs to be stressed is that the Chinese conception of Da Qin was confused from the outset with the ancient mythological notions about the far west.... Attempts to identify them [terms such as Da Qin] with actual western places are obviously futile."