After doing one part of it I wanted to group up for the second part, however I noticed that I could queue them both up again.
Currently I've killed King Mechagon 4 times in one day, getting loot everytime.
So to test my theory I decided to do The Nexus HC in Northrend and also Freehold HC through "specific dungeon" tab.
I got a daily reset lock on them both but not on either of the Operation: Mechagon parts.
I'll provide 2 screenshots before and after I've done Operation Mechagon.
This one showing before.
and after.
As you can see there is no lockout for Operation Mechagon on HC, specifically queued through "Specific Dungeons".
A great way for farming the pets and mounts that drop from these dungeons.