Boruch Sheim Kevod or Yehei Shemei Rabbah Mevorach?

I was putting on my tefillin shel rosh and was about to say boruch sheim kevod after the brachah of al mitzvas tefillin, as is the custom among Ashkenazim.

Right as I was about to recite boruch sheim kevod, the tzibbur was answering Yehei Shemei Rabbah Mevorach. I recently saw that the Targum Yerushlami (Bereishis 49:1) states that the translation of Yehei Shemei Rabbah Mevorach is boruch sheim kevod.

My question is if I may simultaneously fulfill my obligation to recite Yehei Shemei Rabbah Mevorach while saying boruch sheim kevod after putting on the tefillin.

Rav Auerbach:
The appropriate response to Kaddish is only Yehei Shemei Rabbah Mevorach.

Even though the targum of boruch sheim kevod is Yehei Shemei Rabbah Mevorach, since the rest of the congregation is answering a different response, if one were to say boruch sheim kevod during Kaddish, it would not be considered responding to Kaddish together with the tzibbur.

In your case, the proper hanhagah would be to answer Yehei Shemei Rabbah Mevorach and then say boruch sheim.