New to Game capture

Hey folks,
I recently got my granddaughter playing Black Ops 6 with me. I would like to capture gameplay footage and Discord audio to share with family and friends.

My system is more powerful, So I thought doin the capture on my system would make the most sense, but it is custom water cooled and mounted to the wall above my desk, making it harder to work on.

My question is this, would a good quality external Capture card be sufficient, or should I go through the headache of installing an internal card? System specs below…

ME

7800X3D

RTX 4090

32gb DDR5

42 inch 4K LG C4 OLED

HER

5800X3D

7800 XT

32gb DDR4

32 inch Asus TUF 1440P/180hz

Thanks for any advice