I wish there was a web site where you can make and share palettes from 2 to 256/65536 colors (including a possibility to use png images containing all the colors in palette), uncompressed image formats (modes) to fit in some graphics memory size (for example 320x200 16 colors fits in 32KB), an adapter containing multiple of them and possibility to make images fitting these modes. Then the purpose of "best palette" is making images/game screenshots for this site. Then this thread could serve as place in which users could choose one of these suggestions.
Okay. This is good, this is more definite.
In that case, it seems to me that you indeed are not really looking for a 'best' palette, but rather, want a collection of different 'good' ones. For example, Arne's 16color and 32color palettes seem like they would fit into your idea here, and so does the EGA 64color palette and even the CPC 27color palette. Correct me if I'm wrong about any of those.
A second point I wanted to raise is: I'm aware that support for palettes larger than 256colors is much rarer than palettes <=256colors. Have you looked into what software can perform the necessary adaptation?
I also wanted to ask if you have tried GrafX2's RGB restriction option? It seems relevant to your broad goal of exploring these restricted palettes.
It's recommended that palette size is a power of two. E.g. if palette uses 10 colors, it either can be increased to 16 colors or requires inefficient modulo operations. The CPC palette can be increased to 32 colors, while filling in gaps (e.g. brown, two more grays, one desaturated color and ramp fill color).