I wouldn't use caulk... you're likely to just make a mess with it. And, it's likely to interfere with the mechanical operation of the lockset. Besides, there's a better solution, which is to focus on the precise area of the problem.
What I would do is look up "Fasteners" or "Nuts and Bolts" in your yellow pages phone directory to find the places that sell stainless steel screws in your area. Buy STAINLESS STEEL screws to replace all the regular steel fasteners that came with your door hardware. In fact, take the new lock down to that place and match up every screw that comes in the box with a stainless steel one.
Since stainless steel won't rust, it won't seize to the ordinary steel hardware, and you should be able to remove it without a problem.
(PS: This is why one should NEVER EVER NEVER use ordinary steel fasteners in a bathroom. ALWAYS ALWAYS ALWAYS change whatever they give you to stainless steel or solid brass. Lots of plumbing hardware is ordinary steel coated with a thin brass coating to make it look like brass, and in my humble opinion, that stuff should be taken off the market. I've even seen ordinary steel toilet flange-to-bowl bolts plated with brass to make them look like brass, and I think someone should be fired for doing that. Take a magnet with you to check. Brass is completely non-magnetic. Stainless steel can be magnetic or non-magnetic, but generally stainless steel fasteners have anywhere from no magnetism to a small amount of magnetism to them. The magnetism will always be noticably less than that of ordinary steel.)
You should also be aware of a product called "antiseize compound". Antiseize compound is really nothing more than a paste made by mixing extremely tiny particles of nickel or copper into ordinary grease. These metal particles lodge themselves between the male and female parts of a screw thread preventing the male and female threads from actually touching each other, and filling the space between them with a viscous grease that prevents water from getting into the space between them. I always use antiseize compound on the inside of any white metal faucet knobs to prevent them from sticking on to the end of the brass faucet spindles. If you don't do that, it can be like fighting with a bear to get the white metal faucet knob off after a few years.
In your case, I'd recommend overkill. Use antiseize compound on your stainless steel screw threads to GUARANTEE that you won't have a repeat performance of the episode you had with your previous steel door hardware.
You'll find that antiseize compound comes with either copper or nickel dust in it.
Nickel has a much higher melting point than copper, and so nickel based antiseize compounds are used for higher temperature applications (such as in gas turbines or where a copper antiseize compound is inappropriate for any other reason). In your case, you'd be fine with copper.
Also, there's no rule that says you can't also put antiseize compound on the female threads of your lock hardware with a toothpick.