According to ''Advanced Grammar In Use" by M.Hewings, the following rule is given:
After some verbs we cannot include an object before a to-infinitive:
He threatened to report their behaviour to the principal.
(NOT He threatened THEM to report their behaviour…)
Can we say:
(a) He proposed US to go to the theatre (Verb + object + to-infinitive).
(b) Sobir promised HIS MOTHER to be at home early (Verb + object + to-infinitive).
We were taught at school that there is an expression “at home”.
For the proof, I have just looked through the book entitled “English Grammar in Use”
by R.Murphy with Roann Altman and I have read the following statement on page 228
You can say be home/stay home with or without at:
We’ll be out during the day but we’ll be (at) home all evening.
I didn’t go out last night. I stayed (at) home.
Those examples are correct, but nevertheless in your example you can’t say “I’ll be at home early”; it has to be “I’ll be home early”. This is because in your case “be home” means “get home”, or “arrive home”, which is a meaning that “be at home” cannot have.