Stephanie, I'm not sure if you are having problems stripping or staining the outside of the glass area but the problem would be the same. The glass area probably has putty around the opening to hold the glass and seal it. This will not take stain. To remove or reglaze the putty, you have to remove the putty and a heat gun is the most efficient for this. It's not necessary to repaint or stain the new putty and your doors will be happy if you don't, but you do want to apply paint or a clear film finish like polyurethane to any wood that is exposed.
For wood filler, Minwax High Performance Wood Filler works well on exterior applications and well as interior. It will not accept stain quite as well a wood but the difference can range from stark to negligeable. You can blend colors with the stain applicators that are like Magic Markers and you should be able to hide much of the difference. You will hear that this filler or that filler will accept stain "just like the wood". Want to see the MANY cans that I have tried where someone said the same thing?
It's really no big trick to use the markers to blend in the filled areas with the wood and it's kind of fun. If you REALLY make a mess, you an remove the marker with paint thinner and start over.