The balance axis or (balance staff), is quite simply the shaft that the balance wheel, the roller table and the hairspring are mounted onto. The two ends of the staff have very small tapered pivots that break easily.
I truly believe that the charges asked for the replacement of the balance staff (3000$), are ridiculous.
Your watch probably has the Vacheron calibre 1003. I had for the same calibre my staff replaced from my master watchmaker in Athens for 100 Euro.
Find out from VC whether $3000 includes servicing the whole watch including the case. If yes, then, I think it is alright.
Also find out whether $3000 includes any hidden charges by the boutique, ie. handling charges or admin charges or any charges of that sort.
Regards
Ling