"the bigger reason" is correct and usable in written English.
You can use it when you need to emphasize a more significant reason for something as opposed to the lesser reasons. For example, "The bigger reason I decided to move was to be closer to my family."
Exact(44)
But the bigger reason is political fear.
(The bigger reason, though, appears to be cost).
"The bigger reason is that we are righting a wrong.
But the bigger reason appears to be that fog-like suspicion.
The bigger reason for choosing to backdate is to get around some bothersome accounting regulations.
But the bigger reason is that scourge of American youth, team sports.
Similar(16)
The biggest reason is shipping.
"That's the biggest reason".
Actually, the biggest reason Obama and co.
The biggest reason is the tax cut".
The biggest reason is the offense.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com