As I said in a following post I did do that with one character. Now Aela is forever standing in the Underforge waiting. I lost one of the more interesting Companion members, my favorite archery training, and can not finish the Companion quest line.
But if your character is that against lycanthropy, why would you even want to look at her again, let alone talk to her, have her teach you, or follow you around? Why would your character want to continue associating with those mongrels once you learned what they are and what they require of you? In all honesty, your character should be thankful they let you out of the Underforge at all once they made the offer.
Some of the comments here about not doing guild quests that don't fit your character are just ridiculous, you people are telling others hey it's fine skip great big chunks of the game, it's fine there's no problem with them forcing you against your RP. You are actually telling people not to play the game!!
Not every character has to do everything, especially if you're role-playing. If you really can't come up with a good reason for your character to go through the guild quesline, there's plenty to do outside of them until you make another character that's less focused on role-play or is less against what happens in the guild.
Take this character a goodie two shoes warrior - no Dark brotherhood (obvious), no Thieves guild (same), no Mages guild (don't cast spells), I can do Companions (Oh wait I have to become a Werewolf so no to this one too), no Vampires (obvious), Dawnguard (maybe the only one, but wait I have to help out Serana when I should kill her), so whats that make it 0/6.
That's up to how good you can role-play. A role-play needn't be a static adventure, "this is my character and he's going to do this and this and this, and nothing that happens in the game will make me deviate". You can of course play that way if you want, but I find I get much more out of role-playing and the game in general if I just let my character's story tell itself.
I make the character, define their personality, wants, and needs, then start the game. Nothing's set in stone, what happens in the game drives what my character decides to do next. A good guy works through the Companions and finds out their secret? Maybe this changes his perception of that secret, thinking it not as bad as it first seemed, or maybe this steels him and causes him to leave, vowing to attack them if he ever sees them alone and strengthening his vow against "evil", or maybe he feels entrapped and begrudgingly goes through with it as it lowers his opinion of those involved (and then may or may not change him in any number of ways as he experiences it first-hand).