All blog data on a page using custom query gives 404 for page 2 and onward

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I initially thought you’re using the code in the question on a Page (a post of the type page), but if you are actually using that code on an archive-based page (e.g. the home page, a category page, etc.), then the 404 error is normal. You can’t paginate secondary/custom query on archive-based pages. Well, not when using the default paged query string in the URL.

So a possible solution is use a Page and make your custom loop in there — basically, the code in the question would work after you made the changes I mentioned in the original answer. I.e.:

  1. Change the base argument in your paginate_links() call:

    // change from this
'base' => str_replace( $big, '%#%.html', esc_url( home_url( '/<slug>/page/' . $big ) ) )
// to this
'base' => str_replace( $big, '%#%', esc_url( get_pagenum_link( $big ) ) )

  2. And in addition to that base, you also need to change the 'total' => 6 to 'total' => $testimonials->max_num_pages. (more details here)

But if you actually wanted to customize the main query, then use the pre_get_posts hook.

So for example, if your home page is set to show the latest posts, but you want to include only posts of the type blog_post, then you can simply add this to your theme’s functions.php file:

add_action( 'pre_get_posts', function ( $query ) {
    if ( ! is_admin() && $query->is_main_query() && is_home() ) {
        $query->set( 'post_type', 'blog_post' );
        $query->set( 'posts_per_page', 2 );
    }
} );

And that’s really all you need. I.e. No need to worry about the pagination because a good theme always includes a working pagination for the main query on the home page and other archive-based pages.