Don't use any rust treatment of any kind on the clock as a whole as it will attack the brass. No home made remedies, colas, vinegars, jellies, fancy proprietary names, chelates etc all contain acid or equivalent.
Once 'let down' the spring will still be under tension in the barrel so still an energy hazard. Once you have got it out of there still don't use a rust treatment. A rusty spring still springs but many rust treatments affect the microcracks in the tensioned side causing failure. You are better off cleaning with a brass brush that is not hard enough to make surface scratches and relubricating.
Clock springs can be broken. The guy on the TV is an idiot. The common problem is that the spring breaks at the arbor fixing point, or that rivet fails, however if you are feeling that it is still under tension then that isn't the problem.