Can a shliach provide the hankerchief for a Kinyan Chalipin?

In Chiddushei R' Shmuel in Kiddushin Siman 3, R' Shmuel points out a machlokes haposkim whether by kinyan chalipin, if a shliach were to do the chalipin for someone, must he use the mishaleach's sudar, or can he use his own. It is a machlokes between the Sma and the Taz. The question is, what is the point of dispute?
R' Shmuel answers that by kinyan kesef, certainly if a shliach does the kinyan he should use the mishaleach's money. After all, the mishaleach is the koneh so his money should do the purchasing. In fact, Rashi seems to state this explicitly, although the Ritva seems to argue.
On the other hand, by regular kinyanim like hagbahah and meshicha, certainly the shliach does the whole maaseh kinyan.
This then is the question by chalipin. Do we say that the hankerchief acts like a payment for the object being acquired, in which case the koneh should supply it. Or, is the hankerchief just a maaseh kinyan, like hagbahah, in which case the shliach can supply his own.