I would distinguish between 1813-1814 and 1815. In 1813-1814, there was a case for sending Davout to retake Hamburg. By then, he and Napoleon had fallen out, but Napoleon knew that Davout's loyalty was absolute. Napoleon needed a subordinate who could act as an independent commander and governor, and never surrender or switch sides under any circumstances. Which is what happened. Although Napoleon was maybe punishing Davout for his conduct in the Russian Campaign, blaming him for the loss of most of Ney's III Corps. See:
http://www.napoleonicsociety.com/englis ... avout.htmlBut 1815 was different, at least according to David Chandler. Chandler concludes that Napoleon left Davout in Paris for no good reason. Chandler, Campaigns of Napoleon (1966), p. 1022:
"Marshal Suchet would have made a far better chief of staff than Soult . . . by appointing possibly his ablest marshal [Davout] as minister of war and governor of the capital, Napoleon sacrificed the talents of an experienced and gifted soldier."
Perhaps Napoleon was again counting on Davout's loyalty and thinking in terms of what would happen if Paris was attacked (remembering Marmont's surrender at Paris in 1814). But surely Napoleon knew that a great victory in Belgium would be far more valuable in securing his rule than a loyal Paris governor. Perhaps Napoleon was jealous of Davout's military reputation and did not want to share the glory from smashing Bluecher and Wellington and reconquering the Low Countries.
Obviously, if Davout had been in command of Napoleon's left wing or right wing in 1815 (instead of Ney or Grouchy), the campaign would have gone differently. Assuming that Bluecher kept his army in being after Ligny, however, I suspect that Wellington would still have fought the French to a draw. Davout would have recommended a flanking maneuver through Braine l'Alleud to turn Wellington's right, but Napoleon would have rejected Davout's suggestion and insisted on a frontal attack, as happened historically. Wellington would likely have retreated in good order, Brussels would have been occupied, and the war would have been prolonged.