It seems to me that a big part of the problem is the large amount of thrust which is why the aircraft is pitching up so much to begin with. Some of the bigger Boeings which are typically on automatics are designed to only add enough thrust for a 2000 fpm climb if the TOGA switch is pushed once. I am not sure about the 737-800.
I remember in the old 727 being empty on a cold day with a 6000 fpm climb with a level off coming up. All was solved by bringing the power back about halfway at least 1500 feet before level off resulting in a nice reduction toward a 1000 fpm climb and level off for the last 1000 feet of climb.
That being said, the flaps had been retracted at this point. But on a manually flown go-around with a high climb rate and a fairly low altitude level off, there can be some conflicting limits. The missed approach altitude is nearing quickly with a high climb rate resulting in a desire to push forward on the controls to avoid passing through it, but there is also a speed limit on the flap setting. Things start happening very fast which can perhaps easily be slowed down significantly by disconnecting the autothrottle and reducing thrust. Perhaps some button pushing can be done with the autothrottle connected but there seems to be a time lag doing this.