It was the Dover road that lay, on a Friday night late in November, before the first of the persons with whom this history has business. The Dover road lay, as to him, beyond the Dover mail, as it lumbered up Shooter’s Hill. He walked up hill in the mire by the side of the mail, as the rest of the passengers did; not because they had the least relish for walking exercise, under the circumstances, but because the hill, and the harness, and the mud, and the mail, were all so heavy, that the horses had three times already come to a stop, besides once drawing the coach across the road, with the mutinous intent of taking it back to Blackheath. Reins and whip and coachman and guard, however, in combination, had read that article of war which forbade a purpose otherwise strongly in favour of the argument, that some brute animals are endued with Reason; and the team had capitulated and returned to their duty.
(SOED) purpose 2 The action or fact of intending to do something; resolution, determination. [since] Middle English
(SOED) resolution IV 13 A formal decision or expression of opinion by a legislative assembly, committee, public meeting etc.; a formulation of this; Also, gen. (now rare), a statement, a decision, a verdict [since] mid 16th century.
The sentence means that the inflexible coachman and guard—humouristic assimilation of lifeless implements to those two individuals with the intent of strengthening the idea of their unswerving deciciveness—, had in such circumstances of having to contend with the stubborness of carriage animals the definite knowledge that no allowance should be tolerated, and that this determination of theirs was of a ruthless sort—here again, this fact being humouristically imparted to the reader, that is by pretending that the coachman and the guard went by the rigid articles of war, in particular that one which says that the verdict that some brute animals have reason was forbidden, a fact which, if accepted, would usher in convincingly the idea of allowing the team to go back the way they came.
( "argument, that some brute animals are endued with Reason" cannot be read as "the argument is that some brute animals are endued with Reason"; a comma after "argument" wouldn't have been used. The reading has to be, apparently "a purpose that some brute animals are endued with Reason". "Argument" has a referent that one must construct in one's mind, I think: it is the argument that it would have been best to listen to the animals and go back to Blackheath.)
Roughly, "purpose" could be rendered by "verdict".