פרשת וארא - Whose staff is it anyways?

In order to give Paroah a warning for the plague of blood, (ז:טו) Moshe is told to meet Paroah at the nile and "וְהַמַּטֶּה אֲשֶׁר נֶהְפַּךְ לְנָחָשׁ תִּקַּח בְּיָדֶךָ," the staff that was turned into a snake you shall take in your hand. The question is, who's staff is it?

A staff was turned into a snake twice, once in front of B'nei Yisroel and once in front of Paroah. It would seem that in front of B'nei Yisroel it was done with Moshe's staff as HaShem had demonstrated in front of the burning bush. From pasuk 9, it appears that the staff used in front of Paroah was Aharon's. Which one, then, is being referred to in this pasuk?

The כלי יקר on pasuk 9 points out that Moshe's staff turned into a "נחש" while Aharon's turned into a "תנין." He then goes on to explain the difference between the two. Since our pasuk reads "וְהַמַּטֶּה אֲשֶׁר נֶהְפַּךְ לְנָחָשׁ" and not "וְהַמַּטֶּה אֲשֶׁר נֶהְפַּך לְתַנִּין," it would seem that the staff being referred to is Moshe's.

The אבן אזרא asserts that even the staff that was used in front of Paroah was Moshe's. According to this, it would seem to leave no doubt that the staff was Moshe's. [It is noteworthy, however, that when the signs are in fact performed in front of B'nei Yisroel, (4:30), it seems to be Aharon who performed them. Why this would be is a question unto itself. But if Aharon was the one who performed them, perhaps it is not so simple that Moshe's staff was used.]

Nevertheless, תרגום (המיוחס ל)יונתן here states outright that it was Aharon's staff to be brought to the nile. In תוספות השלם, an interesting reasoning for this is brought. Moshe's staff had the שם השם, HaShem's name, etched on it. Rashi here tells us that Paroah was at the Nile bank to relieve himself. Therefore, taking Moshe's staff with HaShem's name on it would have been like taking a sefer into the bathroom. So, it had to be Aharon's staff, one on a slightly lower level of קדושה that was brought to the nile.