Yes, you may pay alternative costs to cast Asmoranomardicadaistinaculdacar, and "without paying its mana cost" is an alternative cost.
118.6. Some objects have no mana cost. This represents an unpayable cost. An ability can also have an unpayable cost if its cost is based on the mana cost of an object with no mana cost. Attempting to cast a spell or activate an ability that has an unpayable cost is a legal action. However, attempting to pay an unpayable cost is an illegal action.
118.6a If an unpayable cost is increased by an effect or an additional cost is imposed, the cost is still unpayable. If an alternative cost is applied to an unpayable cost, including an effect that allows a player to cast a spell without paying its mana cost, the alternative cost may be paid.
Note that "As long as you've discarded a card this turn you may pay [B/R] to cast this spell." also represents an alternative cost, so that's already a hint regarding alternative costs.