How can I tell if a contractor is trying to scam me by charging more after starting a job?
First, find out what the law is in your jurisdiction concerning hidden wires.  It may be that removal is not required. If removal is required, or if you want to have it done for safety reasons, option A is to call another contractor or handyman service and ask for a quote.  If the job can be done cheaper by someone else, it is ok to bring that person in for that purpose.  

If you want the present contractor to do the job, option B is to say "I'll pay $300 after we sign a contract change document, and if you want to quit rather than do the job for my price, I can't stop you."

Option C is to pay the $600 provided that a contract change document is signed.  I wouldn't choose this option without knowing that the removal is legally required. 

You can find out whether removal is required by law in several ways.  First, ask your contractor to tell you what law requires it, then look it up.  Or you can ask any other licensed contractor.  Or you can visit the county engineer (or whatever that department is called where you live) and ask.  Another good question for the engineering department would be whether it would be acceptable to leave the wire where it is if both ends are properly terminated.