If you wish to cast the exiled card, you do so as part of the resolution of Transcendent Dragon’s last ability. You can’t wait to cast it later. Timing permissions based on the card’s type are ignored.

If you cast a spell “without paying its mana cost,” you can’t choose to cast it for any alternative costs. You can, however, pay additional costs, such as kicker costs. If the spell has any mandatory additional costs those must be paid to cast the spell.

If a spell has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.