"Between" meaning "against"
Fowler reads
The regular or ‘weak’ form thrived has won the centuries-old battle between irregular or ‘strong’ throve as past tense and thriven as past participle.
Is this usage of between grammatically correct?
Yes, it's grammatical. No, it's not semantically correct. The battle was between the weak form and the strong forms, while the sentence states the battle was between the two strong forms. You could use against, as you suggest, or reword the entire sentence.