I am not convinced of that - Berndotte's best position would have been to fight along with Davout - and that is the crunch. Napoléon completly failed to take into account that Davout would run into the main army (as for the Prussians pop up at Belle Alliance) and therefore failed to give those wonderfull assez claire instructions. All what he needed to do would be to just issue an order to Bernadotte to support Davout in case he is confronted by the enemy and disregard Napoléon completly. Such an order however was - according to the knowledge - Napoléon had - extremley unlikley there he was convinced to face a big Prussian army.
And then let us look how the battle at Jena devloped, it moved to the west to the last attack / stand of Rüchel.
When do you think would have been an approbriate time for Bernadotte to be at Apolda - at noon?