The Medrash Tankhuma (Balaq 1) says:

… וְאִם תֹּאמַר מִפְּנֵי מָה הִשְׁרָה הַקָּבָּ”ה שְׁכִינָתוֹ עַל גּוֹי רָשָׁע? כְּדֵי שֶׁלֹּא יִהְיֶה פִתְחוֹן פֶּה לָאֻמּוֹת לוֹמַר, אִלּו הָיוּ לָנוּ נְבִיאִים, חָזַרְנוּ לְמוּטָב, הֶעֱמִיד לָהֶם נְבִיאִים וְהֵם פָּרְצוּ גֶדֶר הָעוֹלָם,…

And if you ask: Why did the Holy One blessed be He, let his Shechinah rest upon so wicked a non-Jew? So that the [other] peoples would have no excuse to say, ‘If we had nevi’im, we would have changed for the better’, He established for then nevi’im. Yet they [these nevi’im] broke down the moral fence of the world…

For that matter, the Sifrei says on the last pasuq of the Torah:

ולא קם נביא עוד בישראל כמשה: בישראל לא קם, אבל באומות העולם קם. ואיזה זה? זה בלעם בן בעור.

And another navi did not arise again in Israel like Moshe: In Israel, [another] one did not arise, but among the nations of the world, one did arise. And who was it? Bil’am ben Be’or.

So here you have a prophet with the abilities of Moshe (whatever they meant by that) and yet he was no paragon of moral virtue. He didn’t teach them how to lift themselves up, but how to corrupt the Jews.

So how does that address the complaint of the nations? They had their navi, but they said it was unfair that they didn’t have nev’i’im to give moral instruction — and Bil’am wasn’t capable of leading them in that way.

Perhaps Bil’am stood for them as an example to teach them just that point. The nations are described as complaining that if they only had a navi they would have been as good as the Jews. But Bil’am was there to show them nevu’ah wasn’t the answer. Even being a navi and having the Shechinah rest upon them is not sufficient to make an ideal person.

The whole detour into telling us about an event in the lives of Balaq and Bil’am is such a departure from the rest of the Torah, it is considered a separate book. “משה כתב ספרו ופרשת בלעם — Moshe wrote his book and Parashas Bil’am” (Bava Basra 14b) So why it it included?

Based on the above suggestion, the section teaches us about the dangers of frumkeit. We can get so caught up in the pursuit of deveiqus, one’s personal relationship with G-d, one can end up as self-centered and honor-seeking as Bil’am. We need to start out pursuing moral and ethical behavior, ehrlachkeit, and then the connection with the Divine can be harnessed to reach those goals.