Can I display the percentage 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.percentage %}{{ calculatedBadgePrice.listPrice.percentage}}%{% else %}{{ price.listPrice.percentage}}%{% endif %}

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