Can I display the absolute discount value of the sale in the badge?

Yes, it is possible with the following code:

{% if product.calculatedPrices and product.calculatedPrices|length != 0 %}{% set calculatedBadgePrice = product.calculatedPrices.first %}{% else %}{% set calculatedBadgePrice = product.calculatedPrice %}{% endif %}{% if calculatedBadgePrice.listPrice.discount %}{{ calculatedBadgePrice.listPrice.discount|currency }}{% elseif price.listPrice.discount %}{{ price.listPrice.discount|currency }}{% endif %}