I don't think you need to pack the space between the edges of the beams. The forces on the bracket are concentrated at the edges; more screws or backing in the middle will do almost nothing.
That depends. The Loctek PSW831 mount I suggested has a wall bracket 66mm x 215mm. This would possibly fit between the side timbers but depending on the recess depth might need some packing to allow the arms to fold parallel to the wall.