Lots of hospitals already use pressure mats to detect occupancy in chairs and beds. These are relatively cheap and easy, although you’d have to be buying in large quantity to get under $50. But you can make your own for a few bucks, and then the only real cost is the cost of the antenna. You can definitely put the pressure sensor under a regular cushion and leave the antenna piece attached to the chair frame…
edited to add Crossposted with @sidjohn1 