I think,as someone previously mentioned the stick could have been an attempt to burn her internally. Perhaps he thought that might help completely burn her body to ashes.
The stick as a way to help burn the body internally and destroy evidence, I see the logic in that.
But (just my rambling thoughts):
Would that be the first thing you think of doing when you've just killed someone and you want to cover it up or even destroy evidence? His DNA would be on other parts of her body too. Why not use petrol or bleach or something? Or even bury the body, so it wouldn't be found? As a way of destroying evidence the stick thing seems too "advanced" or "sophisticated", if you know what I mean.
If he wanted the fire to destroy evidence, he probably didn't want the fire to attract too much attention either, so no petrol. And burying the body would maybe take too much time and he was afraid of being seen.
They reported that it was likely an accelerant was used (a lighter?) but that it was difficult to tell because so much time had passed before they discovered the body and no petrol was found, did I remember that correctly? So now I'm wondering just how he thought he could burn a body with a few branches and a lighter... It was said he watched CSI, so he would know about DNA etc, but wouldn't he also know that it wasn't easy to burn a body? Maybe he'd seen something on the telly about a funeral pyre and thought it would be an easy thing to do (especially because he used to make bonfires in the Thicket). Or he thought that was what you did when people died, you have them cremated. Whatever it was, it just feels so clumsy to me.
The body was somewhat hidden if it wasn't found until the 21st... Could he have gone back several times to try and burn the body some more when he went out "searching" for Louise, but he realised it didn't work and gave up, or he couldn't go back after he'd been arrested for kidnapping?
If the main purpose of the stick was to destroy internal evidence, you'd think he'd have set fire to that particular stick first (or to that stick too), wouldn't you? To make sure it caught fire? Was there evidence of fire on that stick? They found his DNA on the stick, so he did't burn it properly, and the "bonfire" was mostly on top of the body, I believe.
For me the stick is (mostly) sexual, as twisted as that is (and even then I can't get my head around it), and the fire a clumsy way to try and cover up what he'd done.
MOO