As Astraport mentions, you need to draw a text field in bitmapData every time you update text using bitmapData.draw() .
If you use textField.getBounds to determine the size of the required bitmap, then the border rectangle will not include an additional size due to the filter (for example, DropShadowFilter sets the side of the text field to certain pixels depending on the "distance" and "blur"). To ensure filters are enabled when drawing a bitmap, you will also need bitmapData.generateFilterRect() to get the correct rect size.
Code snippet (untested, but general idea):
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