I think this highly depends on how the fingerprint search works. This answer is based on the assumption that you use the hardware based fingerprint reader in a table device.
As I see it right now there could be two possible options:
Method 1: Press fingerprint-field first
- The user presses the button (no matter where it is positioned)
- The search box is opened and the user can scan a finger print
- The fingerprint is scanned and the search results are shown
Method 2: Scan fingerprint first
- The fingerprint is scanned and the search results are shown
Obviously, the second method is much shorter and saves interaction time. You should consider there is that the user should know that the fingerprint scanner can be used. A way to do this could be by having the button available as in your 3rd proposal. If the user taps this, a simple popup can be shown that indicates how the system should be used.
Downside there is that the user can accidentally press the fingerprint reader, causing the system to search directly.
My suggestion would be to test this with your users.