In my Opinion it takes one-two three to retrim the Rudder. I did it quite a few times in a hurry when after a Single engine Go, i got the second engine back.
@ His dudeness: A fully loaded loaded non raisbeck B200 climbs acc. to my Manual (BB1716) with 3,8% @ 106KT
Flaps Appr. thats 400ft/min. flaps up its 5,5% @ 121KT so around 700ft/min.
Somewhere in the Flight Safety Training Manual is written: Gear down costs you app. 450ft/min climbrate - Flaps App around 100ft/min - Flaps full app 500ft/min
Add the Sideslip and it wont climb with the Gear down, "if there is a Problem with one Engine".
With 2 Donkeys Running there should Show around 2400ft/min or correcteted for the Overweight some 2200ft/min.
The whole TO Looks in the Video similar to an Engine out, not like a Sideslip, that wont kill 2400ft/min of Climbrate.
Lets say it that way: I have seen People Operate with 13000+lb on 800m Rwys in similar Weather. And i know, with 2 Engines it will work. And Every "Experienced" KA Pilot is watching and working the Ball during the whole flight.
And by the way, i was CRI/TRI on the King Air until i decided to not revalidate it.