Your friend's correction was accurate. A native speaker would say
All he does is watch TV.
Your proposed reversal of complement and subject is also correct: it would indeed be expressed as
Watching TV is all he does.
"To watch TV is all he does" is grammatically correct, but you would almost never hear it in normal speech.
Unfortunately, there is no universal rule governing this usage. However, "does" is almost always followed by the bare infinitive in everyday usages like this one, even when another verb (like "to be" in your example) comes between them.
A thread at the Wordreference.com forum discusses this usage.
The omission of "to" before a complemental infinitive ("to watch" in your example") is expected when the finite element of the first verb is a form of "do" or a modal verb with "do" as dependent:
All he does is watch TV.
All he did was watch TV.
All he could do was watch TV.
The omission of "to" is optional when the finite element of the first verb is a non-modal verb with "to do" as its complement:
All he wants to do is (to) watch TV.
Like many things in English, this initially confusing construction will become more clear as you use the language in everyday conversation and reading.
Best Answer
Notice that it is grammatical (and perhaps more idiomatic) to say "We'd recommend you book your flight early."
Thus, you have an option to use a bare infinitive in both cases. There is no general rule saying that passive infinitives must be bare although in this specific case I agree that you cannot idiomatically just plunk a "to" in front of the passive infinitive without other changes to the sentence.