I'd like to not pick it up - how should I go about doing this?
Well, if your mind is already made up then don't pick it up. Are you just looking for affirmation for your idea?
If your mind isn't made up then keep reading.
The invoice might not be unreasonable but the fact that they didn't consult you first is what's unprofessional.
A simple "Hey, we're estimating $500+ to get your bike working" would have went a long way. Tangentially, you should have expressed a limit at which they must call you: "Hey if this looks like more than $200 then please call me."
It sounds like you gave them a bike and expressed "Please get this working."
You're both at fault so sorting out this matter is a question of interpersonal relation more-so than one of legality. I mean, if they did a good job then would you consider doing business with the shop in the future? Don't burn your bridges so hastily.
Your starting line of defense is to physically go to the shop and say:
Hey, could we discuss this bill? I never received a call telling me that it was getting this high. I thought this was going to be a ~$200 job and this is considerably more than I was anticipating.
In no way am I trying to scam your shop but at this price I would have just bought a better bike. Honestly, I'd be willing to let you keep the bike for $100, the price which I paid on Craigslist.