Could TCAS be used as a backup source for position, attitude or air data? Given the number of incidents related to freezing pitot tubes, is there any current/planned implementation to infer aircraft position/speed/attitude/bearing based on TCAS data as a last resort data point? If not, is this at least doable?

Even if TCAS gives only relative data, one may correlate the data at the last moment before pitot data are lost to assist in dead reckoning.

Edit: I am not sure if my question has been misunderstood, or that I am misinterpreting the answers. The question is more like inferring current aircraft's data from transponders of surrounding aircrafts. Let's suppose at time $t_1$ I have planes A, B, and C within my TCAS range. I will possess information regarding their bearing, distance, speed, and altitude. At time $t_2$, even without my own altimeter, I can at least infer my own vertical speed using trigonometry. A, B, C are like ocean currents, a background moving together with me. If I know enough information about them, depending on what TCAS's underlying data stream provides, I can probably infer my own position, altitude, speed, and heading. From my limited understanding, TCAS bearing, distance, and speed are relative to me. However if I have not mistaken, despite what is shown on the screen, altitude is absolute measurement from the other planes' altimeter. One side fixed on a similar triangle, you'll get the rest.
Can TCAS assist in dead reckoning?

First, understand that TCAS runs off secondary (transponder) radar data. Essentially your aircraft with TCAS is like ATC radar, interrogating other transponders, getting their pressure altitude and getting their bearing from your aircraft.  That data combined is processed to create threat advisories.

If displayed on a moving map, the moving map is typically fed positional data for your aircraft from GPS, and the TCAS alert data is overlayed onto the map.

So to address your question, TCAS only provides data about transponders relative to your aircraft.  TCAS does not determine where you are, or where you are going. (Depending upon your display it might appear that way, but that is not part of the TCAS system.)

So the answer to your question is no, TCAS does not effectively create a secondary navigation capability.

Addendum #1
TCAS can only sense altitude through the host aircraft's static port, so if it is iced over, TCAS has invalid altitude information.

TCAS does not sense other air data, such as pitot pressure information (airspeed).

TCAS does not know the host aircraft's position.  Any positional information in TCAS is derived from pressure altitude information the host and all interrogated transponders.  If the host static port is obscured, the information on threats will be invalid, because TCAS displays relative to the host pressure altitude.

In other words, there is no absolute positional information, and all TCAS information is relative to the current host aircraft, and derived from the secondary beacons of interrogated aircraft transponders.