Bar Kerisus for Kesivas HaGet 3

We previously posted on the issue of Bar Kerisus in Kesivas HaGet here and here. We will now finish this topic (at least for now) with this third, and final, installment.
We stated previously that Rashi holds that one needs to be a bar kerisus to write a get based on the drasha of v'kasav v'nasan. We noted that this drasha doesn't seem to appear anywhere, and we suggested that the source of it is an expanded view of the drasha of ukshartem uchsavtem. Let us now elaborate on that point.
The Gemara in Gittin 45b states a drasha of ukshartem uchsavtem, that one who can "lay" tefillin can write tefillin. The idea is that a goy, eved, woman, or koton are not allowed to write STa"M (sefer torah, tefillin and mezuzos) because they are not "b'toras keshira". Tosafos over there (d"h kol) brings Rabbeinu Tam who says that likewise a woman may not bind the lulav or make tzitzis because she is not commanded in these mitzvos. Tosafos dispute Rabbeinu Tam. They bring a proof from the building of a sukkah, that is kosher even with a goy (sukkas ganbach kesheira) , even though the goy is not commanded in sukkah. Thus, the dispute is whether or not there is a general rule in kol hatorah kulah that to create a cheftza shel mitzvah one must be commanded in that mitzva. Rashi would seem to follow along the lines of Rabbeinu Tam. He holds that even by get, to create the cheftza of a get one must be a bar nesinas get. The drasha of vkasav vnasan is simply an expansion of ukshartem uchsavtem. This follows Rabbeinu Tam's expanded view of this halacha.
What does Rabbeinu Tam do with the building of a Sukkah? Take a look at the sefer B'ikvei Hatzon where R' Schachter brings the Mishkanos Yaakov that Rabbeinu Tam's shittah is only by the creation of a cheftza shel mitzvah that requires lishma. By the building of a sukkah there is no halacha that one must build the sukkah lishma, therefore anyone can build it. (This answer is logical because there has to be some line drawn as to what is considered the creation of a cheftza shel mitzvah. Is planting the tree for the lulav also prohibited to be done by a goy? Obviously not. The Mishkanos Yaakov is saying the line is drawn with lishma.) So essentially, Rabbeinu Tam's shitah is that someone who is not commanded in a mitzvah also doesn't have the power to create that cheftza shel mitzvah lishma. This understanding will fit perfectly with our previous understanding in Rashi and the Rambam. Namely, that when the gemara says a goy can't write a get because he lacks lishma, it really means that he lacks the ability halachically to create a challos of lishma in the get. This was R' Chaim's understanding. Acc. to this, the reason why a goy lacks this ability is because he is not commanded in the laws of gittin.
The question now swings back to tosafos. Tosafos proved that one need not be a bar krisus to write a get from bris milah. We asked, what is the comparison? By milah, one is not creating a cheftza shel mitzvah, he is doing the actual maaseh mitzvah! The proof we brought for Rashi is far superior because the creation of ST"aM is a cheftza shel mitzvah, a hechsher, just like get.
To further strengthen this point take a look at the Bais Yosef in Siman 14 that a woman may lay tefillin on a man (who is incapable of doing it himself) because it is the actual mitzvah and not just a hechsher. So what is the proof from milah?
On Daf 32b in Gittin it appears from Tosafos d"h hasam that one can be mevatel a get. The question is what kind of ridiculous idea is that, could one be mevatel a sefer torah after it was written? How could you undo it being a kosher get? It appears from Tosafos that the writing of a get is not called the "gmar hadavar". It is actually the beginning of the actual maaseh gerushin. See Eretz Hatzvi siman 24 who brings this from R' Soloveitchik. The idea is that the initial writing of the get lishma is the beginning of the gerushin process. Rashi there seems to dispute this point. Tosafos, however, works perfectly in bringing his proof from milah, both are considered the actual maaseh, and not mere hechsherim.
In the end, Rashi and the Rambam hold that the halacha of bar kerisus in kesivas haget and writing a get lishma are all one halacha. Also, they both hold that the din of lishma is part of a process of creating the cheftza shel mitzvah (which must be done by a bar hachi). As icing on the cake tosafos on 9b in gittin state clearly that lishma and bar kerisus are two separate halachos. The halacha of lishma is not related to the creation of a cheftza shel mitzvah, but to the beginning of the gerushin process. Thus, they prove from milah that one needn't not be a bar hachi when doing the actual maaseh mitzvah.

To summarize the main points of dispute:

Rashi and the Rambam: the din of lishma and the din of being a bar kerisus are intertwined. Writing a get lishma is a halacha in the creation of the cheftza of the get. Only a bar kerisus can create a cheftza of a get because only he has the power to infuse lishma into the get. Lishma doesn't merely mean having a certain kavana. It means having the halachic ability to create a cheftza shel get. Only a bar kerisus can do that. This is true by all cheftza shel mitzvah. If the creation of the cheftza shel mitzvah needs lishma, it means that only a bar hachi can create it. It's all one halacha.
Tosafos: the halacha of lishma has nothing to do with bar kerisus. Lishma is just a simple halacha in the writing of a get that it must be done with a certain intention. It is not creating a cheftza shel get by infusing the lishma into the get. Not only that, on the contrary! After a get is written it is not a "finished" cheftza at all. Rather, the writing of the get is viewed as the first step of the actual maaseh gerushin. Like bris milah, the writing of the get can even be performed by someone who is not a bar hachi. Lishma is not related at all to the halachos of being a bar hachi and thus has nothing to do with the "conditions" of the creation of a cheftza shel mitzvah. Rather, it is simply a din in the gerushin process as to what type of kavana needs to be had.

Note: There is a beauty to tosafos' shittah. Because he holds that writing the get is the first step in the gerushin process it is a chiddush that this doesn't need to be done by the husband himself. Rather, as long as the kavana is there it's fine. It is then logical to say that what advantage would being a bar kerisus be? After all, the husband is not doing it anyway. In Rashi, the writing of the get is a separate process from the giving of the get. It makes sense then, to add a requirement that to "create" a get one must be a bar kerisus.