Custom Search on Custom Page while preserving the main search function

It turned out, that the following solved my issue:

function search_rewrite_rules()
{
    global $wp_rewrite;
    $wp_rewrite->search_base="Search Product";
}
add_action('init', 'search_rewrite_rules');

This hook moves the search_base to the given page, which enables everything I wanted to achieve.

Credit goes to the guy in this query:
Custom slug in front of search URL