It's important to remember that for the arm-Tefillin the four sections are written on one piece of Klaf, while for the head-Tefillin they are written on four separate pieces of Klaf and placed in four separate boxes.
The natural question to ask about the head-Tefillin becomes: do the four sections get inserted from right to left from the perspective of the one wearing the Tefillin or do they get inserted from right to left from the perspective of the one looking at the front of the Tefillin. (This is like how stage-right is house-left.) The Talmud (Menachot 34b) actually quotes different sources that phrase the proper order from different perspectives.
But for the arm-Tefillin where it's all on one piece of Klaf and there are no alternative perspectives, of course you'd write it from right to left like any other Hebrew writing. After all neither side denied that the sections are supposed to be right to left in whatever perspective.
From a Rabbenu-Tam/Raavad perspective, the same argument applies: in the head Tefillin the "correct" perspective for reading is from the outside in, so you put section 1 and 3 in an outer box and then 2 and 4 respectively in the box just inward from it. From one side you encounter 1 then 2, and from the other side you encounter 3 then 4. But again for the arm-Tefillin there is only one reasonable perspective to choose: reading in order as written on the single sheet. No one reads from the beginning and end of a single sheet inwards.
The Semak (153) indeed writes that no one argues against Rashi about how the arm-Tefillin is written.
I'm certain there are opinions that recommend being strict to write the Klaf for an arm-Tefillin in the same order as the sections of a paired Rabbenu-Tam/Raavad/Shimmusha-Rabba head-Tefillin are placed in their boxes, but you asked why some would be lenient regarding the arm-Tefillin and I'd expect this to be why.