I put the PGNS and AGS calculations for the theta angle (R3 of V83 in the PGNS, address 277 in the AGS) in a MATLAB script and let it plot the difference. This is using the Apollo 11 attitude of the LM when they were doing the ORDEAL initialization, still docked, about 300° yaw angle and close to 0° in roll. Their pitch angle was 200°, but I let it plot the theta angle as a function of IMU pitch angle.
So there is a difference of up to 60°, as much as the yaw angle. At 0° yaw there is no difference. There is a jump of 120° at the points where the AGS theta is 90° and 270°. This seems to be due to a difference in the definition of the angle. For V83 if the vehicle is pointing above the local horizontal the angle is always between 0° and 180°, otherwise it's between 180° and 360°. The AGS defines the angle as the "in-plane angle between Z body axis and local horizontal". For the PGNS the angle isn't in-plane. So I think it's simply a difference in definition of the angle, not a bug in the V83 code or so. I do think that the AGS definition of the angle is more useful for the ORDEAL initialization. So if you happen to have an AGS onboard, using that instead of V83 is correct. In the CSM you should definitely be close to 0° yaw angle or you get a random error that can be as large as your yaw angle.
I think the theta in V83 wasn't really added for the ORDEAL. Instead it shows the elevation angle of a target you are pointing to. That is how V83 was used during the Apollo 7 rendezvous. You point the CSM (or LM) at your target and V83 gives you the elevation angle of that target. And that works without the AGC having a valid state vector for your target vehicle, e.g. if your sextant has failed and you can't do any marks. From the Apollo 7 rendezvous procedures, optics failure backup technique:
The method for determining the TPI maneuver will be as follows:
After NSR the SCS pulse mode will be selected and the commander will control attitude to maintain wings level and boresight the COAS on the SIVB. The event timer will be set to 52 minutes and holding, awaiting the start of the TPI backup sequence. The CSM pilot will monitor elevation angle from V83 and when 20.0 degrees is reached, the event timer will be started counting up. This will establish the time of TPI and the measurements 5 and 8 minutes prior to the burn.