There is already a Greek word, ready at hand, that Paul could have used had he wanted to say Junia was "well known to" the apostles. γινώσκω "to be known"...and the emphatic form, ἐπιγινώσκω "to be well known".
What BDAG and LSJ are telling us is that if Paul wanted to communicate that Junia was well known to the apostles, then using ἐπίσημος virtually guaranteed that they would misunderstand him. What he said in Greek was that she was an outstanding member of the group of apostles...and this is how native speakers (such as Origen and John Chrysostom) understood him.
But, you say...how can I trust that the big lexicons got it right? After all, there is the ESV. Surely they knew what they were doing, yes?
Happily, the tool one needs to evaluate whether the big lexicons are right or not is ready at hand: the Thesaurus Linguae Graecae (TLG).
The TLG is a database that contains 100% of the Greek writings from 600 BC top 500 AD, and most of the writings from AD 500 to 1453. It will allow you to search through the 110 million words that comprise every book, letter, or shopping list ever composed in Ancient Greek, and in seconds find every instance of the word ἐπίσημος.
You can then download the surrounding context, so you can see how the word is used.
Although you can do your basic search in a couple of hours, it will take you some months to translate your way through all the wrap-around text that surrounds each instance of ἐπίσημος. You will need to do this to sort out the context, and what the author was trying to say with the word.
This is indeed what the editors of the LSJ and BDAG have already done for us...but there is no need to simply trust that they are right. You can verify it for yourself.
Of course, the lexicon makers are aware that people will want to verify their conclusions...so they include the key references they used to come up with their definitions. Most of the text NOT in blue are references. If you purchase a copy of BDAG, you can see what all the abbreviations mean, and you can track down their references.
Once you verify all of these references from BDAG...and perhaps find a few more on your own from the TLG, you will be in a position to say, "No...BDAG was wrong and the translators of the ESV got it right."
Once you document a range of instances where the best possible meaning of the word ἐπίσημος is "to be known by", please do drop me a line. I shall be interested to hear what you came up with.