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If you copy the layer of subtitles into a movieclip, that when you do a test movie the movieclip plays along with the right timing, you can then give that movieclip a drop shadow filter.
Setting up the timing might be tricky, but you could change the movieclip on stage to be a Graphic, so that you can scrub your main timeline and see the text changing, then when you're happy change it back to movieclip, and add the drop shadow filter.
Is this an AS3 or Canvas document? Animate in AS3 mode is perfectly happy to apply filter effects to static text objects. In Canvas mode you can apply filters to static text objects by wrapping them in a movieclip, then applying the filter to the movieclip. However, you should be very careful about using static text in Canvas mode, because it works by breaking every single letter down into vector outlines. This massively inflates the file size, and renders much more slowly.
If this is AS3, I'd recommend using glow instead of drop shadow. With a low Blur (~3) and a very high Strength (~2000%), at Medium quality, you get something very close to an outline effect, which is ideal for subtitles. Unfortunately this doesn't work in Canvas mode because Strength is apparently clamped to 100% when you run it, even though it will look correct in the IDE.