I managed to create a tiny module where I show both prices (with and without tax) using:
$priceWithTax = Tools::displayPrice($product->getPrice(true, null, 6, null, false, false), $this->context->currency);
$priceWithoutTax = Tools::displayPrice($product->getPrice(false, null, 6, null, false, false), $this->context->currency);
Just need to find out how to display the discounted prices with and without tax