Is there a source for this, and, if so, what's the reason?
I don't know that it is necessarily a Chassidish custom, but it is an Israeli one. Some sources include Orchas Mishpat 1. Yabia Omer 3 Even HaEzer 13, Trumas HaDeshen 2:232, Rashba on Bava Basra 175a and Nahar Mitzrayim 192b.
According to Rav Ovadiah Yosef who sums up most of these sources in his responsa in Yabia Omer the reason to to halakhically strengthen the grooms responsibility to pay his Ketubah including the added sum, as well as to make the contract legally binding in a civil court.
As far as the bride signing, I have not heard of that.