And what is he gonna do there? Try to get a laugh out of the broody Wardens?
As Varric once lampshaded, Hawke's the sort of walking weirdness magnet who cannot go to the beach without an armada of angry demon pirates showing up. Sending them to the headquarters of an Order who tend to deal with weirdness and disasters on a regular basis, probably would lead to some interesting shenanigans taking place.
Besides, for a non-Warden, Hawke seems to play Six Degrees of Kevin Bacon when it comes to potential connections to them.
Hawke has a sibling potentially in the Wardens, potentially fought in the Battle of Ostagar (if warrior/rogue), potentially is related to the Hero of Ferelden (if a human mage was the Warden), potentially met Nathaniel and potentially recovered some of Avernus' research for them. Canon connections that happen regardless of class/worldstate include, their father Malcolm Hawke being hired by Wardens to reinforce Corypheus' seals, Hawke being lured to the same prison decades later, having had Anders as a companion who was a rogue Warden, having encountered Wardens several times in Kirkwall, in addition to having made contact and begun working with the Warden-ally inbetween the events of DA2 and Inquisition.
As we can see, they have a history with Wardens that would make sending them to Weisshaupt make sense, even if only to answer questions regarding Anders, Corypheus or the events that transpired in Inquisition such as the Battle of Adamant?