Is it possible to customize the layout of IVS server-side composition?

Yes, Amazon IVS server-side composition does offer some layout customization options, though they are limited to two specific layout types: grid and picture-in-picture (PiP).

For the grid layout (which is the default), you can customize several aspects:

• You can set a featured participant using the featuredParticipantAttribute
• Adjust spacing between participant tiles with the gridGap parameter
• Control whether to omit participants with stopped video using omitStoppedVideo
• Customize participant ordering with participantOrderAttribute
• Set the aspect ratio of video tiles with videoAspectRatio (options include AUTO, VIDEO (16:9), SQUARE (1:1), and PORTRAIT (3:4))
• Define how video content fits within tiles using videoFillMode (FILL, COVER, or CONTAIN)

For the picture-in-picture layout, there are also configuration options available through the PipConfiguration object.

However, it's important to note that there is currently no option for fully custom layouts like placing two hosts in specific positions or adding background images as scenes. Amazon has noted that visual improvements to layouts can be introduced at any time, so it's recommended not to depend on specific features of the current layouts.

If you need highly personalized views with custom elements like overlays or specific positioning that aren't compatible with server-side composition's options, client-side composition would be the more appropriate choice as it offers greater layout flexibility.
Sources
Overview of IVS Server-Side Composition - Amazon IVS
GridConfiguration - Amazon IVS
IVS Release Notes | Real-Time Streaming - Amazon IVS
LayoutConfiguration - Amazon IVS
Broadcasting a Stage: Client-Side versus Server-Side Composition - Amazon IVS

You need to talk to your local aws representative to discuss what in particular is missing.