On a touch and go landing, part of the drill is to first partially open the thrust levers to keep the engine spooled up to around 60% N1, select Flap 15, use the stab trim to set take off trim setting (usually about five units) and rotate at Vref.
Having conducted many touch and go landings in the real 737-200, it is common to see the PF holding back stick to prevent the nose-wheel from hitting hard as it lowers itself.
At the same time, if the PNF is hitting his own stab trim switch to reset to five units from back trim of around eight units or even more, and the PF is holding back stick to make gentle nose wheel contact with the runway, often we see the stab trim not operating because the PF is holding the stick back while the PNF is trimming forward and this opposite movement causes the stab trim to stop working until the PF releases the back pressure on his own control column.