Adam. Anyone with a shade of sense can see why the idea is ridiculous. As its goes Its strikingly similar to Mike's theory of Police shooting Sheila in the kitchen then carrying her body upstairs into the main bedroom 
You are the one closely associated with Mike's and Lookout's theories as you beleive Bamber is innocent. Or rather in you're case 'may not be guilty', despite you're 'forensic evidence breakthrough' apparently being a game changer.
How do you think Bamber got Sheila into the main bedroom ?
There is no reason why she would wake. Thread created. The twins didn't.
If she did wake she would have gone to and stayed with the twins.
If she did wake and walk into the main bedroom, she would have June's blood all over her hands and feet.
If Bamber walked her in at gunpoint to the main bedroom, she would have June's blood all over her feet. And as Sami and lots of other posters said, she would have also resisted.
I would have no trouble carrying somone of Sheila's size who is asleep, a few feet. Even if she started waking there is no way she would have realised what was going on until it was much too late. People can take minutes to wake up from sleeping. You are obviously physically smaller than me or Bamber and believe it would be difficult.