Wolf:
This is actually an interesting machlokes between “Litvaks” and “Chassidim” regarding putting on Rabbainu Tam Tefillin.
Chassidim hold one should put on Rabbainu Tam to be Yotze his shita. The Shulcha Aruch agrees that a “chassid” should wear both pairs.
The Mishna Berura says it is Gaiva for someone who is not a Chassid to put on Rabbainu Tam Tefilin. (34:3)
However, no one seems to be Machmir for the Shita that Tefilin can not have sirtut, as is the Shita of Rav Simcha and the Mahari Abuhav in the Tur/Bach.
Oh well.
Akiva: See my explanation for his shita above. It is not R’ Chaim, it is the Mishna Berura.