It looks like you're already listening for Ctrl+F keypresses and suppressing the default behavior. This is as far as you can go.
In other words, the answer to your question is quite simply NO.
The find-in-page search bar is part of the browser chrome; you do not have any way of accessing it through the DOM or otherwise. Browsers do not provide any kind of API that allows a page to control the find-in-page functionality in any way. You cannot close a find bar that is already open. You cannot cause it to be opened. You cannot read or control what is in the search input box.
You may be able to write an extension that controls the find-in-page functionality in some way, but obviously you'd have to convince users to install it and you'd need one for each browser you're targeting.
Quite frankly, any time I see someone asking to disable or interfere with default browser behavior, it is usually a misguided, bad idea. You might think your search functionality is super awesome (after all, you wrote it), but you might be limiting your users in some unexpected way. (I can't say for sure without seeing your exact use case, but this is a good general rule.)
If you're trying to defeat standard browser functionality, you're probably doing it wrong.