My Japanese dictionary says that はるばる is an adverb and sometimes is used with と. So you are right — the existence of と doesn’t change the grammatical role of はるばる. In fact, はるばる and はるばると are pretty much the same in terms of both usage and meaning.
As background, sometimes adverbs in Japanese are followed by と or に. (See https://kotobank.jp/word/副詞-123983; the 語形的分類 section in ニッポニカ explains this.)
Whether you add と or not depends on each person.
The following is just my opinion as a native speaker of Japanese. This is totally not a linguistic explanation and my major is not linguistics.
Ending sentences with an adverb without と or に seems unusual (though of course there are exceptions, but I think most of them are in the narration of TV shows or literature).
So without a と or に following the adverb at the end of the sentence, it makes me feel like the sentence has ended suddenly. Adding と at the end of the sentence tells readers that there is something omitted at the end of the sentence, for example “やってきた”.
The と helps me recognize that this sentence includes an omitted phrase at the end, and as a result I don’t feel that it ends suddenly, making it feel more natural (especially in a language like Japanese where omitting things is common).