## Why
Sizing, rendering, and cursor placement can query the same chat widget layout
multiple times in one frame, repeatedly measuring active transcript cells.
## What changed
- Build one chat widget renderable tree per frame and reuse it for sizing,
rendering, and cursor placement.
- Cache each flex child's desired height by width for the lifetime of that tree.
- Reuse the bottom pane's renderable directly instead of forwarding each
renderable operation through a wrapper.
## Testing
- Verify flex layouts measure a child once across frame passes and remeasure it
when the width changes.
- Verify a chat widget frame measures its active transcript cell once.
GitOrigin-RevId: 5ad1a6711f4011c699b5d002b13dc3319cb4db8e