According to Ai, "No, the Sony FDR AX53 camcorder does not have a built-in video adapter. It features an HDMI micro terminal for outputting video and audio, but you would need a separate HDMI capture card to use it as a webcam or connect it to a computer."
Your Cam Link should have one end connected to your camera's HDMI port* and the other to your computer via USB 3.0.
Test an mp4 that has the issue. Download this free utility and reencode the file. Launch it, then click Open Video File. Select the file. After it loads, click the Start Encode button. The default result is a 1080 file.
Does the reencoded version also have the problem?