[Note: It should be "I didn't DO it" (not TO it)]
[Note: At least in American English, it sounds a bit unnatural to end the sentence with "do." We would be more likely to say "I didn't do it as he'd asked us to." and leave the "do it" implied. Yes, that ends the sentence with a preposition - most American English speakers don't care. Or, if you are going to include the "do" write out the whole thing: "I didn't do it as he'd asked us to do it." This emphasizes that you actually did the job, but not in the specific manner that it was asked for.
"in the way" and "the way" are equivalent and mean that I did not complete the order to his specifications; I did something, but not what he asked.
"I didn't do it as he'd asked us to do" could mean the same thing, but could also mean that I simply didn't do it at all. The meaning is more vague and more dependent on context. "As" is a slippery word, prone to multiple meanings. There is even the unlikely possibility that you are saying you would have done it for anyone else but him, but because he asked you refused. "I didn't do it, as HE'd asked us to do."