Back in 1982 & 1986 I sent birth announcements to the White House for both my sons. I did get a reply (Ronald Reagan)for both of them sending congrats on the new baby.
For my first son, I also sent an invite to attend his 1st birthday party and also got a letter back saying President Reagan's schedule didn't allow for him to attend but wished him a happy birthday.
All 3 of them were "signed" by the President.
So... I know back then they did reply but who knows now days.
Crossing my fingers for you though!