I hear what you are saying, but as a local I tend to think it is unlikely - but not impossible that he ended up in one of those. I grew up less than a mile from the camp, I have stupidly walked home after a night out in town before, if he went the routes I think he would have gone, I can't see him going near the water...I really can't IMHO.
The Lark is not near to the road and Livermere Lake is set back from the road by a long way. There is no reason to walk through ampton/livermere park. It's off route for him and absolutely pitch dark. You just wouldn't attempt it at night with no light. There is only really one point where the track goes near the lake, over the bridge. That lake and land near it has shoots and gamekeepers on it, it's shooting season, I would imagine that it's been well gone over.
I'm not trying to be argumentative, just trying to give some local insight.