She told me a secret. - Same structure. Unfortunately, English has no distinction between dative and accusative, but syntax dictates that in "She told me a secret", the "me" is dative. And likewise, in "she lent me a car", "me" is dative, not accusative. When we switch the word order, we need to add "to" before "me".
Dictionaries show the same word order for "lend" that is used in this exercice: https://dictionary.cambridge.org/dictionary/english/lend - e.g. "I've consented to lend her my car."
But I agree that the other word order with "...her new car to me" should be added as an option. As for "better grammar", nope, it's equivalent.