The verb آمن has more than one preposition
Hypothesis examined: the verb آمن would always be construed with ب, never with another preposition. This is false: the corpus also uses it with لِ, and the change of preposition changes the meaning.
With ب, the verb expresses adherence to what is believed: believing in God, in the Meeting, in what was sent down.
With لِ, it expresses trust granted to a person who speaks: "Lut gave him credit" (29:26), about Ibrahim. Verse 9:61 places both constructions side by side: "he believes in God, gives credit to the believers". Yusuf's brothers tell their father: "But you will not give us credit, even if we speak the truth" (12:17). And before Fir'awn: "None gave credit to Moussa except a lineage of his people" (10:83).
So the government is not fixed; what is fixed is the correspondence between preposition and meaning. That is the subject of “Believing "in" and giving credit "to"”, already fixed.
In the text
وَيُؤْمِنُ لِلْمُؤْمِنِينَفَـَٔامَنَ لَهُۥ لُوطٌۭوَمَآ أَنتَ بِمُؤْمِنٍۢ لَّنَا