Could fit with the toolbag theory then. She's locked out, he brings and shows her his tools (as opposed to tool), says he can help her break in? Pops them in footwell for easy access and to keep evidence of his false narrative clearly in front of her for reassurance.
Parks in lane round the back of the house, after a show of breaking in, he shows her his tool (screams) and she tries to get away (banging gate).
At this stage he has to have subdued/silenced her v rapidly or else surely we'd have more audio witnesses. Does he swing a hammer at her in panic, she is knocked unconscious or worse, and he bundles her back into the car again? Then...?
Drawback of theory - am not aware of a huge forensics focus on the lane. If police suspected this was a scene of struggle, surely they'd be combing it like the park?
Back to the drawing board.