That is nothing that an FBX can do this way. You need to convert the animation into PLA. (Point Level Animation)
This can be done with “Cappucino” (Shift+C, then type the name, but not in the correct Italian way, like you would when searching the help, then it is Cappuccino), or if there are only a few key positions, manually with the PLA animation option.
Once the PLA information is stored in the Object, switch off any deformer that leads to the animation, and check if the animation is as expected.
Now, export and use the PLA option to create an FBX file and a PLA folder. Never separate these two!
With that, you can re-import to check if it is working.
The FBX settings need the PLA enabled
If all is fine, you can check with other apps.
Here is a tested Round trip file
All the best