I checked the Correspondance there is nothing in it regarding a request.

Still in my view - this would not matter whatsoever, at the 14th of October, when Davout - as well as Bernadotte set their soldiers in motion, both of them (clearly indicated by the order of march by Davout) did not reckon with a sudden major fight, even less with the Prussian main army.
According to their information, submitted by the Etat - Major - Lannes faced a large Prussian Army of about 50.000 and more at the west of him.
When Napoleon first got Davout's reprot about Eckardstberg (as the battle was named firstly by Dabout) - Napoleon could not believe it and did believe it only after he visited the battle field.
And Bernadotte, in what even more akward position would he be - in case there was a detailed report of Davout requesting help. In case Napoleon did not believe Davout did fight the Prussian main army, how can we demand from Bernadotte that he would be better informed and more reasonable?
I dare say, not even Napoleon would have supported Davout because he believed that he was facing the Prussian main Army and in case Davout would have sent a report to Napoleon, he might have accused thim to have double vision.