Tosfos in Chullin on 106b d''h "vechol davar etc." quotes Rav Avraham Ben Harav Moshe. I am wondering whether this is the Rambam's son as this is the same name.
The Tosafot on Chullin are based off of tosafot composed by a student of R. Aharon of Regensburg, who was himself a student of R. Yitzhak of Dampierre (see here, based on Urbach's Ba'alei Tosafot). Thus, it seems eminently likely that the R. Avraham ben R. Moshe you see mentioned is R. Avraham ben R. Moshe of Regensburg. Indeed, Urbach (p665) identifies him as coming from the school of R. Yitzhak.
Urbach mentions R. Avraham ben haRambam three times in his Ba'alei Tosafot, and none of the mentions are to inclusions in our Tosafot.