Eggs on Pesach

It is customary to start the meal by the Pesach seder with an egg (רמ"א סי' תע"ו ס"ב). What is the reason for this minhag?
According to the א"ת ב"ש calculations of the Jewish calendar, the first night of Pesach always falls out on the same night as Tishah Be'Av (the first letter "א" corresponds to the first night of Pesach which matches with "ת" which represents "Tishah Be'Av" - שו"ע סי' תכ"ח ס"ג). Thus, to match the theme of this mournful day, we eat an egg which is the symbol of mourning (as we do on Tish Be'Av). This reason would lead us to eat the egg only on the first night; another reason that we are mournful over the fact that we do not have the korban Pesach today gives reason to eat the egg both nights of the seder.
רמ"א סי' תע"ו ס"ב ובח"י שם סק"ו
ועי' במשב"ז שם סק"ג שמביא טעם אחר והוא משום שאברהם אבינו מת בע"פ ולכן מתאבלים ע"ז
ועי' עוד באשל אברהם שם סק"ד שמבאר מדוע מתאבלים על א"א דוקא בליל הסדר