Josephus transliterates the name as Ἀσαμωναίος. The transliterated form ω corresponds to long o (see Brønno, "Some nominal types in the Septuagint" in Classica et Mediaevalia 3 and Studien über Hebräische Morphologie und Vokalismus auf Grundlage der Mercatischen Fragmente der zweiten Kolumne der Hexapla des Origines). For example, יוֹנָתָן is transliterated as Ἰωνάθην. It would appear, therefore, that Josephus used the form commonly used today.
Nevertheless, old mss agree with u in the name. MS Kaufman A50 (Italy[?], c. 11th cent.), widely considered to be the best extant Mishnah ms, has חַשְׁמוּנַּיִי (e.g. Middot 1:5). Though it is hard to see, it appears that MS Parma 3173 (1073 CE) agrees with this vocalization. The eleventh century MS Bibliotheque Nationale 649 also has this form, but without the dagesh and with the ending אי-.
Thus, the early mss attest to the form with shuruq. Perhaps there two traditions for the pronunciation.
EDIT: From the Cairo Genizah:
T-S AS 105.17 (1r): חַשְׁמוּנַּאי