According to the beis shammai a ben (ches) [Tes] chai is considered a mais so what would be if you shecht it on shabbos? Also from where do we know that a live animal can't become tumai, is it a posuk or sevarah?
(a) Rebbi Yossi ha'Glili and Beis Shammai merely suggest that a ben Peku'ah would become Tamei while still alive. They do not necessarily mean that a ben Peku'ah is considered to be dead (after all, one is punished for Harba'ah and Kilayim of a ben Peku'ah, 74b, see Insights to 74:4). What they mean is that the ben Peku'ah is considered to be a food product, since it is ready to be eaten without further preparation. There is no reason to believe that it would be permitted to kill it on Shabbos.
(b) The Rabbanan's source for saying that a ben Peku'ah is Tahor while it is alive is not entirely clear. From TOSFOS (81b end of DH Ho'il) it seems that it is based not on a Pasuk but on a logical proof: "It's liveliness prevents it from becoming Tahor."