This took a fair amount of time, but it worked! Thank you!
For anyone else doing something similar or wondering how I did it:
I got the AnimationMetadataSection from the TextureAtlasSprite via reflection
I calculate the current frame and amount between the current frame, and next frame to use as alpha, each tick using the AnimationMetadataSection
I have my model create pieces using the texture coords for the current frame and for the next frame
When my model renders, it renders the current frame's pieces and then renders the next frame's pieces, on top, with the alpha