I am having difficulty envisioning the scene where the key got pushed into a pocket. If practical, try to get it out with a shop vac. If the nozzle itself does not work, use some thinner vinyl tubing attached to the nozzle with duct tape. How thin? I don’t know. Pick up some short lengths of different diameters and try them out.
Is it the fob with the battery that is signaling the car or the key (no battery) itself. If the latter it must gets its power by induction by an outside field and transmit with that power. Theoretically, there is a way to find it with an inducer and a receiver of the correct frequencies. It would take some serious thought, knowledge and equipment to find it in the manner that you suggest. A smart ham radio operator that is versed in antenna design might be a good source of help.