"Get intoxicated" vs. "become intoxicated"
I think I would prefer become in that sentence.
With the insignificant amount of alcohol it is impossible to become intoxicated.
As I commented, they are practically synonymous, but get may have the connotation of a deliberate action for the purpose of intoxication and become something more passive. There is an idiomatic phrase "get drunk" which does seem to connote intention; to "become drunk" almost sounds accidental. But that's a subjective view and probably unjustified.