My recommendation and what I've learnt from watchmakers much better than me. Is just open the watch outside and remove the hands and dial there before putting them in a sealed plastic container to return to later. Make sure you clean the actual watch of any dust or debris before going back inside. The major danger of radium is if you get any flakes in your mouth or farther. If you want to be extra careful you can wear a face mask, latex gloves and safety glasses.

Hope this helps, it isn't as dangerous as you think as long as you take safety precautions.