From Vachee, 58-59:

'As to Bernadotte, Murat was in a position to set him in the right direction, since they were working together. But, owing to the transmutation of the army, Augereau at Kahla was, every but as much as Lannes at Jena, expsed to an encounter with the enemy, and, as he himself wrote to Lannes on October 13: 'The Emperor had given him no other orders than to move to Kahla.' We can detect in this phrase Augereau's regret that he had not received instructions as to his role. However, like a good comrade, he subordinated his movements to those of Lannes, with whom he was materially and morally in harmony. One cannot say the same of Bernadotte and Davout on the night of October 13-14. The conflict which arose between these two marshals and which resulted in Bernadotte's regrettable inaction in the battles of the 14th, would doubtless have been avoided had Bernadotte been more completely informted of the idea of the Emperor's maneuver.'

Berthier sent complimentary orders to Murat and Bernadotte on 12 October at 0400. That being the case and the comment to Bernadotte to 'Arrange with him [Murat] regarding your march' seems to me to inform Bernadotte what was going on based on the latest information at Imperial Headquarters (see Vachee, pages 60-61.)

Davout was sent an order at 0500 12 October ordering him to Naumburg, to which Murat and Bernadotte had already been directed (Vachee, 64).

Again from Vachee, 259-260:

'On returning to his bivouac, [Napoleon] thought of Davout. At ten o'clock in the evening [13 October] he sent him a fresh order, the tenor of which was as follows: The Emperor has recognized a Prussian Army which stretches a league away before and on the heights of Jena as far as Weimar. He proposes to attack it on the morrow. He orders Marshal Davout to proceed to Apolda in order to fall on the rear of that army. He leaves the Marshal the choice of his route, provided he takes part in the fight.' The Chief of the Staff added: 'If Marshal Bernadotte is with you, you can march together, but the Emperor hopes that he will be in the position which he pointed out to him at Dornburg.' No other order was to be sent to Davout and Bernadotte before October 15 at 5 am.'

'Davout, nevertheless, acted with a vigor which we admire, but Bernadotte profited by this silence to remain inactive on October 14 during the two battles which were being waged on his right and left.'

Seems to me Vachee is damning Bernadotte inaction here, not being kind. So your previous reference to Vachee is just a little puzzling.

From Maude, 175-176:

'No account of these two battles can be complete without an allusion to the extraordinary failure of the I Corps to participate in either.'

Bernadotte was with Murat and Davout at Naumburg when the orders of Berthier, dated 3 pm, 13th, were received, by which the former was ordered to Dornburg, and the latter told to stand fast unless heavy firing was heard from Jena until later instructions should reach him. As his Corps had been marching all day, he took shelter behind the promise of later orders and halted him men, along the road to Dornburg, on the right bank of the Saale with the tail of the column in Naumburg, where he himself passed the night with Davout. There seems no doubt that in the general disposition he had been overlooked, for when the orders dated 10 pm, Jena, were despatched to DAvout, (received at 3 am) directing him to march on Apolda, the Major General added: 'If Marshal Davout is still with you, you can march together, but the Emperor hopes he will be in the position indicated to him at Dornburg.' Now no position had been indicated to Bernadotte, hence he concluded that some order had missed him, and though this note to Davout must have effectively covered his responsibility had he acted on it, he decided to march to Dornburg. Had he acted with energy on his arrival at Dornburg his presence there would have been more effective than his presence at Aeurstadt, hence he does not seem to be seriously to blame. Unfortunately, for his reputation, however, he did not act with anything approaching energy, but carried out his march so tardily that he failed to reach either field in time to render assistance.'

'By the time he had reached his assigned station, the fog had cleared, and it appears that he was fully informed as to the progress of the fight at Auerstadt, but no specific order from the Emperor reached him, and he determined to march on Apolda, gains as good a direction as he could have chosen, but his march was conducted with such slowness that he only reached that spot at 4 pm, having taken nearly six hours to cover eight miles, and nothing in the nature of defiles to be overcome exists or existed to justify the delay.'

Nothing in the above two volumes exonerates Bernadotte for his actions, or lack of them, on 14 October.

And the material already posted by Napoleon, Savary, and Rapp certainly demonstrates guilt on Bernadotte's part of lack of support to Davout on the 14th. The addendum to the 2200 14 October order by Berthier certainly communicates the intent to Bernadotte that if he is with Davout, which he was, that he was to march with him and not leave him. Bernadotte's corps was on the road to Dornburg but it was halted with the end of the column still in Naumburg. It would have taken time to change direction, but that would have brought him into support Davout at least at the pursuit of the Prussians which would have made a pursuit from the battlefield more effective and would have inflicted more damage on the main Prussian army.

Finally, the fact that Davout was in contact with the Prussians before Bernadotte turned southwards was a definite 'indicator' that going to Dornburg was overcome by events.

Bernadotte contributed nothing, and it definitely appears it was deliberate. If it wasn't, his march to Dornburg and then to Apolda would have been faster because of the situation.

Sincerely,
K