Dogs! Elizabeth Goudge treated family dogs as attentively
as she treated the human characters in her books.

The dog called Mouse is a little gray Cairn terrier.
I think a female? She lives with the Eliot family.
But she shows up only in book 3 of the Eliot group of books.
The first two Eliot books feature two aging dogs
called Pooh-Bah and The Bastard.

The Eliot family includes the bomber pilot, and you are right,
he is in book 3.
"The Heart of the Family" is the title of Book 3 as you posted.
Actually he is an Eliot, the pilot, and as an Eliot he is in all three books;
but in The Bird in the Tree, the first of the Eliot books,
he has yet to serve in the armed forces.

Now, then: the nasty teacher.
This sounds like a different Elizabeth Goudge title,
"The Rosemary Tree,"
about a private school for girls
whose headmistress dies after an illness;
one of the nasty teachers is confronted by the parent of
two sister students.
Because the parent is the local vicar,
he not only confronts the nasty teacher
but forces a conversion experience and a penance!