Problems improving the inquiry speed of Bluetooth HC-05
You can't change how fast Bluetooth devices report their presence.

Bluetooth works the opposite way that you expect. An inquiry doesn't actively request what devices are there. Instead it sits and listens passively for devices to announce that they are there.

If a device enters the target zone and then leaves again before it has broadcast it's "I am here" signal there is nothing you can do.

That is why inquiries take so long. They wait enough time for devices nearby to announce that they are here.

I am not familiar with that specific module, so I don't know if it is always listening in the background and maintaining a list, or just listening during and enquiry, but either way the onus is on the remote device to announce that it is there.