Oh GS,
I note your mother phoned you, and not your brother to invite herself along!!
The key fact here is that it was your brother who arranged the dinner for your father's birthday, and, therefore, regarding etiquette your mother needs to phone your brother for an invite.
If it were me (she says boldly sitting on the outside looking in) then I would phone mother, and gently but firmly, say that she needs to phone her son, your brother, directly regarding her attendance at the dinner, as he has arranged the evening.
The birthday card could be mailed.
It really is inappropriate for her to attend, and will most likely create unnecessary tension for everyone.
Is there still time to phone her and highlight the dilemma that you are now placed in?
Love & a Hug,
Leah