The encounter you are looking for
Arg’il interrupts Cassie’s coronation and later rides Frostmaw in the Windforce encounter. His orders appear in the combat log. There is no separate HP target or second Arg’il damage phase documented in the reference.
How to progress
Defeat Trinity Council, continue through the Seventh Sanctum, then complete Frostmaw’s battle while keeping Cassie alive. The conclusion involving Arg’il is part of the story sequence, not another card-build check.