The CXADC sounds interesting but unfortunately drops raw samples upon seeing input sync pulses unless you greatly decrease signal amplitude (which may be fine given it's 1-bit). I wonder if inverting the composite signal would bypass this. Alternatively the Domesday Duplicator is more expensive and may not be intended for composite signals. TBH I wonder if an ARM microcontroller could PLL to the sync pulses, sample the active signal digitally, and output a HDMI signal for a capture card (transmitting to a PC via the Pi Pico's 12 Mbit/s USB is marginal).
While researching LumaCode (similarly based on extracting digital signals from retro hardware) I came across RGBtoHDMI and https://lukazi.blogspot.com/2022/02/apple-ii-video-to-hdmi-u.... I have not read the article but it may be useful.