Studio chat completion: side-thread composer parity #2165

Parent: #2160. Design reference: stacklok/mecatl-prototypes#39 (open prototype).

Outcome

Side-thread composers get the same completion, chips, and capability gating as the main composer.

Scope

• Use the shared editor and both menus in the side-thread panel.
• Keep side-thread draft, send, and Escape behavior unchanged.

Acceptance

• @ and / behave the same in a side thread and in the main chat, including capability gating.
• Existing side-thread tests pass; new tests cover completion there. task studio:check passes.

Delivery

Bounded; reuse, not a second implementation.

Depends on: #2162, #2163, #2164.

Fully or partially written by an AI agent.