What about the footwear he was wearing at the time?
Killer Luke Mitchell went from wearing dark footwear to light footwear
From memory, AB initially described (from the book, IB) a youth wearing 'big boots' along with 'army clothing'. I've always presumed this to be a pair of chunky black Doc Marten boots. It would be handy if AB's full transcript was uploaded to read what she said exactly in regard to LM's footwear that day, though I'm sure if she'd seen white boots she would have said so and SL would've mentioned it in her book ('IB'). Again, it would be helpful if the full transcripts were available of all the eyewitnesses who saw LM between 1650 - 1820 (1820 being aporoximately the time he went back home to begin disposal and change clothes and footwear before meeting the boys in the abbey @ 1930) to read what they all said in relation to LM's clothing & footwear that evening.
I don't think the boots were destroyed in the burner (too messy and risky), but discarded somewhere well away from the house. SM was (deliberately or coincidentally?) out and about in Bathgate seeing a friend that night, so it's possible he dumped boots and clothing belonging to his wee brother there, off the beaten path. Or CM got rid of them; as per the episode with the knife purchased on December '03, she was able to hide things successfully -- even from a professional search team.