Someone in a comment pointed out a sicha of the Lubavitcher Rebbe (Nitzvaim-VaYeilech 5749 found in Sefer haSichos for that year, vol2) dealing with the omission of shofar on Shabbos. I am not sure I can do justice to the Rebbe's sicha in particular, but it's worth discussing more generally some of the approaches in chassidus to this issue. As we have discussed in previous posts (here and here), the Achronim all are troubled by the omission of the mitzvah of shofar on Shabbos. How can we surrender a mitzvah which is so integral to the theme of malchus (tamlichuni aleichem... b'mah? B'shofar) and the theme of kaparah (see Ramban, VaYikra 23:24) which define the very essence of Rosh HaShana because of the remote chance that someone may carry a shofar on Shabbos? Rosh HaShana without shofar is almost like Sukkos without a sukkah -- something of the very fabric of the holiday is missing.
The Rebbe quotes the Ba'al haTanya's answer: Tekiyas shofar is a means to awaken our desire to accept the dominion of Hashem, his malchus. On Shabbos we do not need an extra signal to inspire that desire; the day of Shabbos itself draws us to accept Hashem as our king.
Why then is shofar blown in the Mikdash even on Shabbos (R"H 29b)? Because a person who is within the Mikdash has the potential to be aroused and inspired to even greater heights than can be reached by the celebration of Shabbos alone.
The Shem m'Shmuel's answer: The gemara explains the bent shape of shofar as a hint to bend oneself in submisiveness to Hashem. The shofar reminds us of the need for humility; it arouses us to tshuvah by reminding us of our shortcomings. The celebration of Shabbos is the antithesis of that idea. On Shabbos we delight in being in Hashem's presence and remind ourselves of the greatness of what it means to be a Jew. There is no place for the trembling wail of a shofar on Shabbos, the day of oneg and happiness.
While normally these contrary emotions of humility and greatness, of fear and love, cannot co-exist, within the walls of the Mikdash one can reach a state of trancendence and find harmony even in opposites. Therefore, within the Mikdash, even on Shabbos the shofar is blown.
The Ba'al haTanya and Shem m'Shmuel are really two sides of the same coin. The appreciation of G-d's greatness goes hand in hand with a recognition of one's own limits. The Ba'al haTanya sees shofar as a reminder of the lofty dominion of G-d and hence unnecessary on Shabbos; the Shem m'Shmuel sees shofar as a reminder of the shortcomings of humanity and hence not fitting the Shabbos spirit.
The L. Rebbe's contribution to this discussion is returning the focus to man's avodah. Both the Ba'al haTanya and the Shem m'Shmuel see Shabbos as the day in which G-d expresses his majesty over the world and draws us to his service. We sit in passive acceptance of that state of affairs, spectators who experience Hashem's malchus, not agents who bring it about (Shabbos is "keviya v'kayma" and does not require a kiddush by Beis Din). But, argues the Rebbe, doesn't Rosh HaShana celebrate the creation of man in particular because man alone is charged with actively revealing G-d's majesty, not just experiencing it? We don't celebrate 25 Elul, the start of creation and revelation of G-d's power in the world, but rather 1 Tishrei, the day of mankind's creation, the day of the revelation of man's power, as man alone can change creation for better or worse by accepting G-d as his master or c"v rejecting him.
The theme the Rebbe develops is that our choice to not blow shofar is itself an active expression of our own agency. Blowing shofar during the week expresses our acceptance of Hashem as king; our choice to cease all activity on Shabbos, including shofar blowing, expresses that same acceptance. Between the lines lies the paradox: blowing shofar and the choice to not blow shofar represents hisbatlus to G-d's dominion, but that hisbatlus is effective only because it is an expression of our independent choice.
The sicha gets into deep territory in explaining why we do blow shofar in Mikdash. In a nutshell, hisbatlus is necessary only as a means to resolve the tension between existance as an independent entity, and G-d, the ultimate negation of anything's independence. The Mikdash transcends that conflict; within its walls reality and G-d can be seen as one and the same. The choice of not-blowing as an act of hisbatlus is therefore unnecessary. (Again, the sicha is pretty complex and I admit I have not done justice to it, especially this point, so please see it inside. The sefer can be found on hebrewbooks.org.)