Was Yehoshua Ben Karcha a member of Beth Hillel or Beth Shammai? I am talking about the Yehoshua Ben Karcha of Mishna Berachot 2:2.
Yehoshua ben Karcha was almost certainly Rabbi Akiva's son. Ben Karcha means son of the bald one which was a code that he was the son of Rabbi Akiva who was indeed bald. This name was utilized to protect Yehoshua's real identity from the Romans to avoid persecution.
We see from Bava Metsiah 84b that Rabbi Yehoshua ben Karcha learned in the beit midrash of Rabban Shimon ben Gamliel in Usha (of the fourth generation). Since the latter is a direct descendent of Hillel, we can reasonably assume that his beit midrash taught according to Beit Hillel. This would make Rabbi Yehoshua ben Karcha a member of Beit Hillel.
More circumstantially, according to Rabbeinu Gershom and Rashi on Bechorot 58a (Rashi s.v. חוץ מן הקרחה הזה), Rabbi Yehoshua ben Karcha was the son of Rabbi Akiva (but Tosafot disagree). We have circumstantial evidence to support Rabbi Akiva being a member of Beit Hillel, see also Tosafot on Bava Metsiah 37b s.v. מודה אני בזה and Wikipedia (not sure what their source is). This is an indicator that Rabbi Yehoshua ben Karcha was also a member of Beit Hillel.