The Elgato capture card should work for 480i(Passthrough and x2[At least])/480p (Passthrough) 240p (Passthrough, 2x, 3x, 5x(if set to 1920×1080, though the capture card couldn’t capture 60fps)) as the Game Capture HD is on the older side and might have issues with non regular resolutions, it might work with the other modes but from what I can tell you would have to use OBS instead of the elgato software.
The Elgato Game Capture supports the following resolutions as far as I can tell:
1080p (simultaneous 60 fps pass-through and 30 fps capture), 1080i, 720p (60 or 30 fps capture), 576p, 576i, 480p, 480i, 288p, 240p
To be honest I wouldn’t trust the ebay capture card very much it doesn’t even list supported resolutions aside from 720p/1080i.
To be honest though it might be on the more expensive side I’d recommend the Avermedia GC573 as it works great with my OSSC.