What grounding should I use for this capacitively coupled input?
Video receivers use an active DC restore circuit during Hsync interval to restore the video white level to 0V.

A less precise method uses video with negative sync tips to restore DC with the series cap and shunt reverse biased diode to clamp the negative sync tip to -0.6V such that back porch DC level is stable.

The more precise active clamp uses and suitable analog switch or FET transistor to short circuit the video during the back porch for ~1us or less with suitable RC time constant from 75 Ohms such that DC is restored after sufficient lines of video in a repetitive pulse clamp process.

You get to choose the C value based on x&lt;1% droop  at the end of the video line from input load current and choose switch resistance &lt;&lt; 75 Ohms to respond as fast as 75C =Tau can respond and result in x% IRE DC offset for white level.