Check the DATA line on the keyboard. Hook up the scope with clips if you can, see if pressing a key gives you all lows or all highs no matter what key you press.
I'm guessing it would be all highs. Which would point to the DATA line being held high by something inside the CDTV as you figured already.
Especially if it's happening between the two keyboards, which work fine otherwise.
There is also a crystal oscillaor in the keyboard, you could check that to make sure the keyboard controller is getting a nice clean regular signal (unlikely to be that, but worth checking,
Now, assuming all the above is correct... to be positive it is the CDTV - repeat the test with the DATA line disconnected from the CDTV, but the clock and power lines connected.
You should see the data line wiggling up and down freely if it really is the CDTV. When different keys are pressed or released.
Finally, check with no power at all on the port. Is the data line shorted to the +5V line? Or does it float around?
If it floats, it's got to be a faulty component. If it's short, then it's a short on the board somewhere rather than a faulty component (harder to find, easier to fix than getting a spare chip).
It could just be a faulty resistor pack inside the CDTV,.. Might not be such a difficult replacement,
To verify that, again with the power off, check continuity between pin 39 on the even CIA and the DATA pin on the connector. then continuity between the CIA chip and the +5V rail.
Both the KClock on pin 40 and DATA on pin 39 of the even CIA are connected to a pull up resistor pack. If that's gone from being a 10K resistor to being close to a dead short (as far as the DATA pin goes) that would cause the fault.
EDIT: Resistor pack is labelled RP6 on the schematic.
EDIT EDIT: When you get to measuring resistance between KDATA pin 39 of the CIA and +5V (should be no lower than 9K Ohms allowing for tolerances) then do take the chip out of the socket. That way no circuits in the chip will interfere with the measurement.
If it is much lower, replacing the resistor pack should fix it. You might have trouble finding one with the right number of connections though.