You can add this PHP Snippet to remove the Cart Button:
remove_action( 'woocommerce_widget_shopping_cart_buttons', 'woocommerce_widget_shopping_cart_button_view_cart', 10 );
NOTE: You will need to clear the cart transients cache ( simple way is to add / remove ) something from the cart to see it update.
Then a little CSS to make the Checkout button fill the width:
.woocommerce.widget_shopping_cart .woocommerce-mini-cart__buttons a {
width: 100%;
}