Add tax free fees to WooCommerce cart programmatic

I try to add fees based on some calculations I do on Woocommerce cart, but I want to exclude it from VAT. This is my code:

function woo_add_cart_fee( $cart ) {
    global $woocommerce; $bookable_total = 0; 

    foreach(WC()->cart->get_cart() as $cart_item_key => $values) { 
        $_product = $values['data'];

        //doing my stuff to calculate $fee variable

    WC()->cart->add_fee( 'Fees: ', $fee, false, '' );
    //WC()->cart->add_fee( 'Fees: ', $fee, true, '' );
    //WC()->cart->add_fee( 'Fees: ', $fee, false, 'zero rate' );
    //WC()->cart->add_fee( 'Fees: ', $fee, true, 'zero rate' );
}

add_action( 'woocommerce_cart_calculate_fees', 'woo_add_cart_fee' );

I have tried all the commented versions and each of them includes a VAT on Fees too.

Any idea how I can achieve it?

(Update): TAX OPTIONS with add_fee() method

IMPORTANT: The fact that TAX is working or not with add_fee() method depends first of your tax settings in woocommerce. As you have not tell in your question what are your TAX settings, It's not possible to help you (Tax settings can be much more different for each e-commerce web site).

For example if you want to use 'zero rate' tax class, but you haven't defined the correct 'zero rate' tax class for the customer country, this will not work if you try to use it with:
WC()->cart->add_fee( 'Fees: ', $fee, true, 'zero rate' ); … depending on your global tax settings.

Here is a screenshot of REAL checkout totals, for 3 items in cart (using the code below):

Class WC_Cart add_fee() method, adds additional fee to the cart.

add_fee( string $name, float $amount, boolean $taxable = false, string $tax_class = ''  )

Parameters:
    $name      Unique name for the fee. Multiple fees of the same name cannot be added.
    $amount    Fee amount.
    $taxable   (default: false) Is the fee taxable?
    $tax_class    (default: '') The tax class for the fee if taxable. A blank string is standard tax class.

Original answer (updated code):

Your main problem is in this line: global $woocommerce, $bookable_total = 0;

As you are using WC()->cart syntax instead of $woocommerce->cart syntax, you don't really need global $woocommerce;.

If you use global $bookable_total = 0; this $bookable_total will be always = 0.
Instead you will use global $bookable_total; without a value to get the value defined outside your function.
But if you want to set the value to zero, in case is not defined outside your function, you will do it this way: woo_add_cart_fee( $bookable_total=0 )

We can defined now $bookable_total variable value outside the function.

This is a working example with your code:

// This variable value is passed to our function
$bookable_total = 1;

function woo_add_cart_fee( $bookable_total = 0 ) {
    global $bookable_total;

    if ( is_admin() && ! defined( 'DOING_AJAX' ) )
        return;

    // just for this example
    $item_count = 0;
    $item_fee = 5;

    // going through each cart items
    foreach( WC()->cart->get_cart() as $values ) {
        $item = $values['data'];
        if ( empty( $item ) )
            break;
        // getting the cart item_id
        $item_id = $item->id;
        $item_count++;
        // your calculations
    }

    // We test $bookable_total value, defined to '1' outside our function
    // and to 'O' if not defined outside (in this case the fee will be '0')
    $fee = $item_count * $bookable_total * $item_fee;

    // add_fee method (TAX will NOT be applied here)
    WC()->cart->add_fee( 'Fees: ', $fee, false );

}
add_action( 'woocommerce_cart_calculate_fees','woo_add_cart_fee' );

This code is tested and it works. It goes on function.php file of your active child theme or theme.

If the $bookable_total variable is not defined outside, the value will be 0.