Show the weight and a message indicating the remaining weight on the WooCommerce cart and checkout.

functions.php

add_filter( 'woocommerce_before_cart', 'display_total_weight_notice' );
add_filter( 'woocommerce_before_checkout_form', 'display_total_weight_notice' );
function display_total_weight_notice( $message ) {
    // DEFINE the allowed weight limit
    $allowed_weight    = 3;
    $cart_total_weight = WC()->cart->get_cart_contents_weight();

    if( cart_total_weight <= $allowed_weight ) :

    wc_print_notice( sprintf(
        __( 'Your order has a total weight of %s. The remaining available weight is %s for the current shipping cost' ),
        '<strong>' . wc_format_weight($cart_total_weight) . '</strong>',
        '<strong>' . wc_format_weight($allowed_weight - $cart_total_weight) . '</strong>'
    ),'notice' );

    endif;
}