Advice for RGB selector with CBT3244AD

Hi,
I'm hoping someone with more knowledge about circuit design can help me figure this out.
I've built a switch box with multiple SCART inputs (and one VGA for Dreamcast) that all output to my RGB-Component converter. The switch box works great except it seems that the bus switch chips don't have enough bandwidth to avoid clipping on loud music or bright white screens. For instance, the music in Super Mario RPG or Super Mario 64 has instantly-noticeable clipping when I plug my SCART cables into the switch box. The Sega Saturn boot screen blacks out when it gets too bright.

I wired up a SCART input directly to my component converter and the audio distortion and video blackouts do not occur when I have consoles hooked up this way. I've read other forums where people say the CBT3244 isn't necessarily designed for analog signals, and that "The power rails to your switch need to be compatible with your signal - that is, you need to have +/- power which is higher than your signal swing, otherwise the signal will clip." The person who wrote that didn't explain how one would go about doing that. Since I don't have any formal electronics training, I don't know how to go about improving this. I thought about getting a different switch chip to try such as the IDT QS4A105G, but I can't tell if that would simply work better for AV applications or if I would run into the same problem (not to mention those chips are tiny). Does anyone have any recommendations? Thanks!