Sinytra Connector requires forge version 47.1.3 on 1.20.1, Cave Biomes requires 47.1.47. Is there any way to lower the needed version further for support?

Yes, Sinytra on 1.20.1 doesn't support newer Forge versions and won't change this due to focusing on newer Minecraft versions,

Not sure I understand, why are you not able to use 47.1.47? Does Sinytra Connector require specifically Forge 47.1.3, without support for newer versions?

What happens if you try to use a newer version of Forge? I just booted up Sinytra Connector on Forge 47.3.0 and it seems to boot up okay

EDIT: well I do see that they explicitly mention using 47.1.3 on their CF page. I'll see if I can lower the required version for Cave Biomes