Answering Amen on Tefilin

Question

I understand there is a machlokes whether the mitzvah of tefilin is 2 separate mitzvos or rather one mitzvah with 2 parts. Thus, in terms of blessings, some Ashkenazim make 2 but say ‘Baruch shem kvod malchuso’ - בָּרוּךְ שֵׁם כְּבוֹד מַלְכוּתוֹ לְעוֹלָם וָעֶד - at the end of the second in case it is actually an unnecessary blessing.

Can/should a person answer amen to the second blessing on tefilin, if it extraneous does this pose a problem?

Answer 1

R Jack Abramowitz at OU answers your question this way

After adjusting the head strap, the person continues with “Baruch sheim k'vod malchuso l’olam vo’ed” (Blessed be the Name of He Whose glorious kingdom lasts forever) because there is some doubt as to the need for a second blessing on tefillin. This doubt also raises the question of whether or not someone who hears the blessing “al mitzvas tefillin” should respond with “Amen,” so the Kitzur Shulchan Aruch recommends that this blessing be recited quietly.

Kaf Hachayim 25:40 writes that a Sephardi who hears the bracha on the Tefillin shel rosh is permitted to answer amen in his head (Halachipedia).

Note that Sefaradim (following Rambam Hilchot Mezuza 4:5) and Chabad say only one blessing.

Answer 2

The Kitzur Orach Chayim 10:4 says that due to this uncertainty (see Orach Chayim 25:5) one should say this blessing in an undertone:

וּמִפְּנֵי שֶׁיֵּשׁ סָפֵק בִּבְרָכָה זֹאת, לָכֵן יֵשׁ גַּם כֵּן סָפֵק אִם הַשּׁוֹמֵעַ יַעֲנֶה אָמֵן אוֹ לֹא (פרמ"ג). עַל כֵּן נִרְאֶה לִי כִּי טוֹב לְבָרֵךְ בְּרָכָה זֹאת בְּלָחַשׁ.

(...) and because there is a doubt regarding this berachah there is consequently also a doubt if one who hears the berachah should answer Amen or not. Therefore, it appears to me that it is preferable to recite this berachah in an undertone.

_{Translation from Sefaria}

R. Shlomo Kluger discusses in detail whether one could respond amen to this blessing in Haelef lekhah Shelomoh. Regarding Sefardi authorities, Kaf haChayim repeats this custom, and if someone else said this blessing aloud, one should say amen silently in his heart (see also Yalkut Yosef 1:28):

ואם ברך בקול רם ושמע חבירו, יש לענות אמן בלבו

However, at DinOnline (based on Chacham Ovadia Yosef) the answer was that Sefardim are not required to respond at all:

ובשו”ת יביע אומר ח”א סי’ כט כתב, שאם השומע הוא ספרדי שמנהגו שלא לברך ברכה זו, לא יענה אמן.