B787 First Officer reported a temporary loss of aircraft control during departure climb in a light aircraft in which CRM miscues resulted in a high workload situation. The crew worked together to regain control safely.

Narrative

The crew departed from ZZZ on Runway XXR full-length after a lengthy delay due to weather (4000 RVR; fog; rain) and multiple aircraft going around. The aircraft was a light TOGW at 435K and Flaps 10. The takeoff sequence through rotation; initial climb; and the ZZZX [departure] climbing LEFT and Right turn was uneventful with the pilot flying (PF) hand flying. Upon rolling out on heading 220 and checking on with departure; ATC issued us direct ZZZZZ and climb to 6000 ft. During the climb and turn sequence with a low TOGW and resulting high climb angle; the Autopilot captured the altitude in the climb (relief pilot later stated he believed as early as 1200 ft.) when ATC issued our continue climb instructions. As the pilot monitoring (PM); I spun in 6000 ft.; confirmed with the PF; but did not push the ALT button; not realizing the VNAV had already sequenced to ALT. The following sequence of events occurred in rapid order: I spun the PF requested HDG bug turn; FMS Direct-to sequencing; and confirmations; during which I was focused on the center nav display and heads down in the FMS; and the relief pilot stated 'We are leveling' followed shortly by 'We captured 2;500.' The PF then directed 'Push the button' which I immediately did. This caused the autothrottles to sequence into THR REF. At that moment; the PF and I both realized the aircraft had previously accelerated and reduced thrust at the flap 10 limit speed as it leveled at 2500 ft. PF directed Flaps 1 (skipping 5) which I executed immediately followed by PF calling for Flaps up. This essentially executed a Flaps 10 to Flaps up retraction sequence. This resulted in a combination of high-thrust setting for an intended climb; flaps tracking from 10 to up; and the PF hand flying the aircraft. During this sequence a slight and unnoticed pitch-down caused a rapid acceleration into the barber-pole overspeed tape. The unnoticed downward nose angle I believe aggravated the already occurring acceleration into the barber pole from the THR REF throttle setting and lack of required corresponding nose up climb. This caused the PF to override the throttles with his hand and pull them back toward idle; (likely increasing the downward pitch moment). The relief pilot then stated very definitely 'We are descending' which had the intended effect of focusing all 3 of us on the pitch attitude of the aircraft. I immediately confirmed the relief pilot's statement by repeating 'We are descending' and the PF immediately began to fly the nose up; arresting the descent. The lowest alt we descended to was 2100 ft. and then began a climb up to 6000 ft. ATC confirmed that we understood our climb instructions to 6000 which I replied that we were currently at 2100 and climbing to 6000. After we confirmed as a crew that we were wings-level climbing away and slowing out of the barberpole; the relief pilot suggested that we engage the Autopilot which the PF did a few moments later. We continued the climb and flight uneventfully.