[noggin]

Does the Nvidia Shield support 10bit output because I have never seen it in the display output options?

The Shield TV supports 8, 10 and 12-bit output I believe. (12-bit is required because at 4:2:2 2160p50 and above 12-bit is used for 10-bit content, as there is no 10-bit 4:2:2 option in the HDMI specs)

I’m trying to do 10-bit 23.976 but I don’t see that option. Only 12-bit

There's no major issues with outputting 8-bit SDR and 10-bit HDR video in 12-bit - you're just padding the LSBs with 0000 or 00 respectively usually (rather than dithering).

4:2:2 subsampling only supports 12-bit depth video at 2160p23.976-60 - there are no 8, 10 or 16-bit depth options - just 12-bit - at that resolution in HDMI 2.0

4:4:4/RGB supports 8,10,12 & 16-bit depth video at 2160p30 and below, but only supports 8-bit at 2160p50 and above.
4:2:2 supports 8,10,12 and 16-bit depth video at 2160p50 and above, but is not part of the spec. for 2160p30 and below.

4:2:2 12-bit is the only subsampling that supports HDR (i.e. >8-bit depth) at all frame rates in 2160p.  

This is why most consumer devices that support UHD/4K - UHD Blu-ray players, Satellite set top boxes etc. - often run at 4:2:2 12-bit output in 2160p modes - so that they can run the same subsampling for all frame rates - whether SDR 8-bit or HDR 10-bit sources, and whether outputting at 23.976fps or 50/59.94fps.

Now if you're trying to fix your output at 2160p23.976 then RGB/4:4:4 options should allow you 8-bit, 10-bit and 12-bit and 4:2:2 should allow you 12-bit only.  However the Shield TV and/or your display and/or your AVR may not support all of these - or it may be that the Shield will still prefer an option that will also work at 2160p50 and above (which after all, Kodi can still request a switch to, even when the system settings are set at one frame rate)