I would like to have a custom button in Shopify that would add certain item to card on click based on a fixed product ID.

I did try to add this code to custom HTML on a custom page in the page editor:
page editor

<button id="custom-add-to-cart" data-product-id="11564478366041">Add to Cart</button>

And this javascript code to a new asset file named custom.js:

document.getElementById('custom-add-to-cart').addEventListener('click', function() {
    var productId = this.getAttribute('data-product-id');
    addToCart(productId, 1); // 1 indicates the quantity

    function addToCart(productId, quantity) {
        fetch('/cart/add.js', {
            method: 'POST',
            headers: {
                'Content-Type': 'application/json',
                'Accept': 'application/json'
            },
            body: JSON.stringify({
                id: productId,
                quantity: quantity
            })
        })
        .then(response => {
            if (!response.ok) {
                throw new Error('Network response was not ok');
            }
            return response.json();
        })
        .then(data => {
            console.log('Product added to cart:', data);
            
        })
        .catch(error => {
            console.error('There was a problem with the fetch operation:', error);
        });
    }
});

And this to the theme.liquid:

<script src="{{ 'custom.js' | asset_url }}"></script>