Windows 7 Pro 64-bit
NVidia GeForce GTX 1060 6gb
Broken: Blender 2.82 & 2.83 (all versions tested)
Lattice modifier with shape keys displayed correctly in viewport (both still frames and animated)
Rendered result for any single frame - using 'Render Image' - is correct and as expected.
When rendering out complete animation - using 'Render Animation' - the lattice deformations are ignored and not rendered.
It does not make a difference if the animation is rendered out to a series of images files or to one single video file.
Please examine attached Blender file.
Note the top of the cube is animated using a lattice modifier with shape keys (growing and shrinking). This is the part of the animation that does not render correctly.