One other option to consider if your multiplexer only supports 256-colours: it’s easy to patch Kakoune to use an approximation from that palette instead of requesting the exact 24-bit colour.
In my local build, I have https://github.com/arachsys/packages/blob/master/kakoune/palette.diff which should apply against both the most recent release and current master. I don’t use this much, but it’s really handy when I find myself accessing one of my machines from an older terminal.
To be honest, I don’t notice the slight colour-infidelity when I turn it on, although it’s just about visible if I open two terminals side-by-side. The original thread has a screenshot.