I'm afraid that the various good reasons for you to rent a car in LIS will not help you much in the discussion with TAP. TAP offered to rebook you into a same day flight so there was no obligation to send you by rail or rental car instead.
As for the afternoon LIS-FAO flight (which most likely have been sold out at the time of rebooking) it is a bit late to become aware of that flight. Did you inquire about any earlier TAP flights to FAO? (as one of the first things I would have investigated this on my own and specifically inquired about this flight when talking to TAP about rebooking). Even if sold out in the morning there would still have been chance that you would have been accepted for the fligh on standby basis in the afternoon.
Sorry, I don't see any succesful outcome of this claim.