In the clause 'He died as he had lived', what is the grammatical function of 'as he had lived'? I know it modifies 'died', and I know 'as' can be used.
He died [as he had lived].
As the Original Poster suspects, the string as he had lived is a Modifier of the verb phrase died. (It's a verb phrase as opposed to merely a verb because it forms the predicate of the clause). Modifiers of verb phrases or clauses are often referred to as 'Adjuncts' (or 'Adverbials').
The string as he had lived is a preposition phrase, where we see the word as heading an equative construction. In old-fashioned grammars, the word as would be considered a subordinating conjunction. In modern grammars it is considered a preposition regardless of whether it is followed by a noun phrase or a clause.