(See part I first!)
The Sha’agas Arye answers his question by explaining the reason we do not assume “stama lishma” by mitzvos like we do by korbanos (see Zevachim 2), meaning absent kavanah to the contrary we assume a person does have intent to be yotzei, is because there is no reason a person must use a specific shofar, lulav, talis, etc. to fulfill his mitzvah – since any mitzvah object can be used at any time to fulfill these obligations, what designates the object in hand at this moment as the one being used for a mitzvah and not practice or some other use? Intent is required.
This logic applies by most mitzvos, but fails when applied to the korban pesach. Since the korban pesach is eaten only by those who have reserved a place in a chaburah beforehand, the specific animal to be eaten is already designated for the mitzvah purpose. Even Reish Lakish who ordinarily requires kavanh by mitzvos to complete the designation as a mitzvah will agree that this requirement is unnecessary when it comes to the korban pesach.
The same logic holds true, argues the Mahartz Chiyus, in our sugya of to’eh b’dvar mitzvah for one who carries a lulav on Shabbos. Since it is the carrying of this specific lulav for the intent of fulfilling a mitzvah which is what exempts the person from bringing a korban chatas, it is obvious that the person wishes that the act of carrying be counted for a mitzvah purpose. In this case the designation l’shem mitzvah can be assumed even without the additional element of kavanah, answering Tosfos’ question.
I don’t know if I will be blogging much over chol hamoed. I barely scratched the surface with a few sukkos topics - so much good stuff out there to learn over Yom Tov – enjoy!