I would unlock the cog with the keys, then I would take down tthe rope from one of the shrines. I would use the canvas to make a basket which I would then tie to the rope. I would throw one end of tthe rope over the domed shrine to make a pulley and tie the other end of the rope to the cog. I would then chuck all the heavy things in the room like the statue and crescent moon in to the canvas basket which would pull down on the rope, turning the cog, with my help as well and hopefully retracting the door bar. I would then use the second key to unlock the door (if necessary) so I can leave.
That's my plan after just seeing the puzzle, would it work? It probably needs a bit more refining but I replied on my phone without access to the description of tthe room.
On second thoughts as above but use the Protestant cross like an alan key, slot it in to the axle holein the cog and use it as an axle. Tie the rope to the cross and as above use the other heavy things to pull the rope and turn the cog.