Your quadsticks have bluetooth. You could set up the bluetooth as a combo mouse + keyboard to the computer and be connected via XBox 360 mode over USB and mouse over Bluetooth. You can even mix mouse/keyboard and controller outputs on the same sheet by selecting "both" for the channel in cell C3 of the configuration sheet.
Switching USB modes will always be slow.
Alternatively, using ViGEmBus to create a virtual XBox controller and leaving the Quadstick in its PS3 or PS4 modes (USB modes 0, 4, or 6 (no-flash)) would give you simultaneous controller + mouse/keyboard access.