R. Kook phrases this query differently while encapsulating the underlining question: Is blowing shofar likened to sacred speech ("dibbur shel kedushah") or is it a mere action/mitzvah. The given ramification is if one may fulfill the aforementioned obligation in an unclean place or in the presence of an uncovered body-part "for which we don't find that it is prohibited in such circumstances, save for Torah and tefilah" (Orach Mishpat §139). Biur Halachah too (§588 s.v. שמע) raises this issue and leaves it unsettled.
In looking for precedent we find in Yerushalmi Demai (1:4) that the deconsecration of Demai in a bathhouse is permitted since no blessing is required. Similarly, Rema (YD §19:1), upholding the opinion of Aggudah, rules that it is permissible to ritually slaughter an animal (i.e. the mitzvah of shechitah) in an unclean setting, though the prescribed blessing should be recited beforehand in a clean place. It would seem from these sources that a mitzvah without speech may be performed in an unclean setting.
R. Ovadia Yosef, in a responsum about tithing Terumah and Ma'asrot in an unclean setting (Yabia Omer vol. 6 YD §29), also marshals the Yerushalmi in Demai and suggests to prove that one may indeed tithe in an unclean setting as well. Additionally, he cites other authorities who permitted various biblical mitzvot to be performed in an unclean setting basing themselves on this Yerushalmi. However, he initially objects to equating a biblical mitzvah (tithing Terumah or, as in our case, listening to shofar) which requires intent (cf. Shulchan Aruch OC §60:4), and forbidden to do so in an unclean place (cf. Ber. 24b), to a non-biblical or only preparatory mitzvah (shechitah, deconsecration of Demai etc.) which, according to some authorities, does not require intent.
Yet, R. Yosef concludes that since many authorities, including some Geonim, maintain that even biblical mitzvot do not require intent one should therefore rely on this opinion and perform a mitzvah in event of extenuating circumstances and will not have another chance at appropriately performing the given mitzvah.