At the current release state 2024.2.2 FlexSim does not provide a tool to export an animated USD stage of a model. There is a way around though by recording the FlexSim model connected to a NVidia Omniverse live session using one of the tools in NVidia’s USD Composer, the Animation Stage Recorder. This is a step-by-step tutorial to guide you through the process. It is assumed, when you are interested in this you already know how to connect FlexSim to a live session in NVidia Omniverse.
This may not be ther perfect way to go, but it worked for me.