Hi,
You can try this option: https://i.imgur.com/QqbhlrH.png
When hitting the return or the magnifier key, the plugin will go to the first matching result.
There is also a feature to scroll and highlight the match, however it may not work in combination with the automatic redirection to the first result, as it is programmed for user interaction: https://i.imgur.com/iHAKppk.png
Clicking a tab requires some custom coding for sure, as it is specific to the exact implementation. It should not be too difficult, a developer should be able to do it for you in a few hours.