Delayed or Asynchronous DML Trigger Execution
As far as I'm aware this is not possible with the triggers feature of Azure / SQL Server. One reason likely being, as you mentioned, it breaks the Consistency property of the ACID principals of a database:

...any data written to the database must be valid according to all
defined rules, including constraints, cascades, triggers, and any
combination thereof. This prevents database corruption by an illegal
transaction...

Triggers are meant to fire transactionally in lockstep with the operation that caused them to fire. Imagine if the row of data that the trigger was dependent on was deleted before the trigger got to execute asynchronously. This could get you into a lot of trouble.
Alternatively, you can have a trigger that simply adds a row to a queue table, and a SQL Agent Job for example, that constantly runs checking that queue table and firing a stored procedure with the logic from your original trigger to emulate an asynchronous event. Perhaps that would improve your workflow?