I exported an FBX in text format and can't find any information about Time-Code.
The manual has no idea, either.
If you know the start and end points, set the project accordingly.
Typically the animation is mapped onto the exact time in your timeline.
At least I have tried all combinations in the past hour, and it works, no matter what.
My comments were only meant to suggest where to focus next. I'm not critiquing. Since I write in a forum, I know we have various experience levels here. Hence, I point out certain things, perhaps obvious to most artists. In the editor view, all might look fine because the refresh happens more than once. During rendering or export, there is only one time to "walk through" all the items in the scene. What is not available will be replaced by the material that might have been from a single or a few frames before; if nothing is there, then there is no reply. This is my current conclusion with this setup I got. To be clear, the results visible in the editor view are very nice.
It sounds like a more significant production, whereby I'm not clear about anything besides this scene you shared. So, my suggestion might be off, but how about Taiao from Insydium? https://insydium.ltd/products/taiao/