At Agincourt, Charles D'Albret's battle plan was thwarted as a direct result of the irrepressible desire the French nobility had to charge headlong into any enemy offering a challenge.
This sentence indicates causal relationship, e.g. cause of someone's plan at agincourt was thwarted because of x reason. So usage of 'because' is preferred.
A. as a direct result of the irrepressible desire the French nobility had to - Modifier error, irrepressible desire should referred to The French and desire is incorrectly used as subject of causal clause.
B. because the French nobaility had an irrepressible desire to - Aha, perfect, elegant and meaningful sentence with correct verb usage. Correct
C. insofar as the French nobility desired to - incorrect usage of insofar against original sentence. Irrepressible word is removed.
D. because the French nobility would have an irrepressible desire to - Incorrect verb, would have, used, which distorts the meaning of the sentence.
E. by the French nobility's desire to irrepressibly - Nonsensicle meaning of the sentence because of usage of possessiveness of the noun, the french nobility's.
Imo. B