A cross-platform way to share a USB device over a network?

If you are open to NOT use a RaspBerry PI and use a ready-made solution:

There exist networked USB hubs. A small device with 1 or more USB ports and a RJ45. Accompanying software is installed on a Windows PC to connect to the hub and have it appear to windows as if it is a USB hub directly attached to the PC.

They are mostly used for cluster environments where a VM (that needs a USB key or USB license dongle) can be moved between several physical nodes. You don’t want to have to move the USB device between the physical nodes when you move the VM.

These networked USB hubs work perfectly fine for this application too.

A very well known brand of these is AnywhereUSB.
(I’m not affiliated with them, just a happy customer of their products.)