You can move the command seat into a storage container but it loses its icon. You can interact with it and later remove it from the container, but while it is in the container it is invisible. You can hover over the inventory and see that it is there from the tooltip pop on mouseover, just not the icon.

Is similar problem with seat from USI Exploration Pack

It's a KSP related bug. You can check this ingame : the command seat tooltip icon are empty too.
I currently use icons generated by KSP.
I'm waiting Squad to fix this issue.

I made some research.
Here is the result:

1. The problem is caused by a shader. Broken shader: KSP/Alpha/Cutoff
2. This shader is used in a few of places here and there but the command seat is the only part that doesn't have other shaders.
3. Another part where the problem is noticeable but not fatal is a launch clamp:
   

My repository: https://github.com/mongoose11235813/CommandSeatIconFix/releases
Download fix: https://github.com/mongoose11235813/CommandSeatIconFix/releases/tag/0.1

I managed to fix it. I'll upload my fix soon:

@mongoose11235813 : Nice one! So the problem is caused by a shader!

As squad didn't seem to fix this issue, I implemented in the KIS iconviewer class the shader change for the command seat icon (this way I didn't touch to squad parts prefab).
Here is the commit : 9d0b150. This will be included in the next version of KIS.
Many many thanks for your help :)

Thank you

Not at all
BTW I'd recommend you a slightly different approach (#80) because some modded parts (e.g. USI rover seat) are using this shader too.