Inserting and Removing the Spring Rod for Toilet Paper Rolls

Is it permissible to insert or remove the spring rod used for toilet paper rolls on Shabbat?

It is not permissible to insert or remove the spring rod used for toilet paper rolls in the bathroom on Shabbat.

It is explained in the Shulchan Aruch (Siman 313) that a window plug may be inserted on Shabbat because it is made to be frequently inserted and removed. However, a rod that is inserted into the wall for locking, since it resembles building more than a window plug, requires modification to be permissible for insertion. According to Rashi (Shabbat 126a), the rod must be suitable for some additional use besides locking. According to Rabbeinu Tam, it is sufficient to designate the rod for locking by an act. The Mishnah Berurah (Siman 313, Se'if Katan 7) rules that only in a case of necessity can one be lenient according to Rabbeinu Tam. According to Rashi, there is no permission to insert and remove the spring rod. According to Rabbeinu Tam, it would seem permissible. However, even according to Rabbeinu Tam, one cannot be lenient with the spring rod because the rod that is permissible is used for locking. The door is sometimes open, sometimes closed, and sometimes the rod is not in the wall at all, so it is permissible. But with the spring rod, which is always closed, and it is removed only to insert the roll and immediately returned to its place, this is also considered part of building according to Rabbeinu Tam.