Fix managed child model picker layering

[leoncheng57]

Summary

• render the model picker above nested managed-child dialogs instead of behind their backdrop
• make the owning modal inert and hidden from assistive technology while the nested picker is active
• contain keyboard focus inside the picker and restore focus/parent accessibility state on close
• verify selecting GPT-5.6 Sol updates the managed-child trigger, ledger provenance, and submitted prompt payload

Verification

• npm run typecheck
• npm test — 510 passed
• npm run build
• targeted managed-child E2E — 8 passed
• full E2E — 275 passed, 1 screenshot-runner skip
• independent code review completed; accessibility finding resolved
• git diff --check

This PR intentionally changes only nested model-picker behavior. Automatic native child delegation is unchanged.