Sweet bike. Remove the rust with oxalic acid and remove the flaking chrome with a brass brush (it won't scratch the remaining chrome).

To use a commercial rechroming place, they will have to strip the whole bike as you thought otherwise they would contaminate their plating tanks.

Get the Copy Chrome Plating Kit from here:
http://www.caswellplating.com/kits/plugnplate.htm


I have never used it personally but I have heard it does a decent job. And for $35, you won't be out a lot if it fails. Please post your results.