I think you should probably forget about software USB solutions at this point; they're apparently "not quite conformant", and increasingly fail to work with newer host-side USB chipsets (USB3, USB-C, etc.) besides, chips with HW USB support have become much more common, much cheaper, and with better library support.
It seems it's not possible to use bidges as HID devices (mouse or keyboard).
Sure it is. You just need a different bridge chip. For example, Here's a HID chip from the people who make the CH340 USB/Seral chip.