Seems possible to me.
You'll need 2 arduino's and 2 bluetooth boards. You could also use cheaper nrf24l01 or other wireless modules as well though.
Instead of using a pressure sensor it may be possible to detect someone using Capacitive sensing
, requiring just a piece of metal and 2 resistors.
With an transistor+ relay you could operate the button.
I'd do my best to hide 'evil' intentions by the way. When the doorbell starts ringing at the moment people sit down, they'll soon know there's a relation between both.
If the bell starts ringing after a random period of sitting and occasionally 10-20 secs just after they left the chair though it will be much harder to detect what's happening.