Should one be particular to wear Rabbeinu Tam before Musaf even if it will cause him to miss Davening Musaf with the Minyan?^[1]

No. One is to Daven Musaf with the Minyan and only afterwards wear Rabbeinu Tam.^[2] However, there are some who are accustomed to wear the Tefillin before Musaf even in such a case.^[3] It is disputed amongst Chabad Rabbanim as to how one should follow.^[4]

[1] Os Chaim Veshalom 25 footnote 20; Piskeiy Teshuvos 25:27

[2] The reason: As Davening with a Minyan is a very important Mitzvah in which one sanctifies Hashem’s name in public, and one may hence even transgress a negative command of the Torah for this purpose, such as to free a slave. [Admur 90:17; unlike Michaber Yoreh Deah 269:79] And it pushes away buying food for Shabbos which is a definite Mitzvah, either Biblical or at least Rabbinical. [Admur 250:3; Unlike Biur Halacha 250] Now, wearing Tefillin of Rabbeinu Tam is only a Mitzvah of Yirei Shamayim, and practically we rule that it can be worn after Musaf, and hence one is not to make the main Mitzvah of a Minyan be differed in face of a secondary Mitzvah of Rabbeinu Tam. Therefore, those Chassidim that place on Rabbeinu Tam even if it causes them to lose the Minyan are to be protested. [Os Chaim Veshalom ibid] Now, although Chassidim and those who follow Kabala follow a specific order of Davening and do not skip this order even in order to Daven with the Minyan, and likewise we allow Davening in length even if it causes one to miss Davening with the Minyan, nevertheless, this is only because the delay is for the sake of a better quality of the Davening. However here, placing Rabbeinu Tam before Musaf is not to give better quality to Musaf but to give better quality to the Tefillin, and as proved from above a Minyan pushes off all Mitzvos.

[3] See Os Chaim Veshalom ibid

[4] Rav Asher Lemel Hakohen of Beitar rules like the Os Chaim that one must Daven with the Minyan and wear the Tefillin afterwards. However, Harav Eliyahu Landa and Rav Groner said one is to first wear the Tefillin and then Daven later Beyechidus. [verified through personal correspondence]