If you cannot get the screw out by conventional means then I would recommend soaking the bridge in a solution of alum (aluminum sulfate – available at any grocery store). Leave it overnight and the steel screw will be dissolved due to electrolytic action. If you want to speed up the process you can heat the solution (simmer). This will not harm the brass bridge or its nickel plating.
This is from JimH on the Timezone website, and I’ve seen many others who’ve used it with success.
Look for alum in the shaving section..it’s the old-school way to stop razor nicks.
For the future , there are two products that you can get at auto parts stores.
Evaporust- non toxic, really cool stuff that dissolves rust off of almost anything, without eating most finishes.
You might try it with an old movement plate to be sure,..but I used it on some plated tools last week and it worked great.
‘PB Blaster”” is known amongst the auto crowd as a great lubricant, and a much better rust releaser than Liquid wrench.
Hope one of these helps Tmac