At first I thought that the answer could be that the Meiri is really a compendium and is just quoting the Ramban; this however didn’t fully satisfy me. I then perused all the manuscripts that relate to this issue and discovered something fascinating. Our passage from Ulla in Meseches Berachos (58a) which states "אמר עולה נקיטינן אין אוכלוסא בבבל. תנא אין אוכלוסא פחותה מששים רבוא" is different from the Paris codex manuscript of Meseches Berachos (Bibliotheque Nationale, Hebrew. MS. 671) where it just says "אמר עולה נקיטינן אין אוכלוסא פחותה מששים רבוא" and omits the mention of Bavel. This version of Ullas statement from the Paris codex seems to be the version that the Meiri had in Meseches Berachos (there are some other Rishonim that had this version as well). I therefore, posit that even though the Meiri had the passage from the Vatican codex of Meseches Shabbos (6a) where Ulla states that there is no reshus harabbim in Bavel he didn’t have the passage from the Paris codex of Meseches Berachos (58a) where Ulla states that there is no אוכלוסא (multitudes) in Bavel. Accordingly, he didn’t know of the connection between the two Gemaras which can be understood as being the source in Shas of the criterion of shishim ribo. Consequentially, the Meiri cites the Ramban’s objection to shishim ribo since there is no mention of it in his manuscript of Shas.
This facsimile above is of Meseches Berachos (58a) from the Paris codex (Bibliotheque Nationale, Hebrew MS. 671) and shows where Ulla’s words (the omission of Bavel) differ from the text in our Gemara.