fix: preserve model context across background agent handoffs (#229)

Fixes #191

This commit ensures that the user's selected model is preserved when
background tasks complete and notify their parent sessions.

Changes:
- Add parentModel field to BackgroundTask and LaunchInput interfaces
- Capture model context when launching background tasks
- Pass model context when notifying parent sessions after task completion

Impact:
- Users with OAuth providers (Google, Anthropic) will now have their
  model selection preserved across background task continuations
- Background agents no longer revert to hardcoded defaults

Co-authored-by: sisyphus-dev-ai <sisyphus-dev-ai@users.noreply.github.com>

src/features/background-agent/manager.ts

@@ -99,6 +99,7 @@ export class BackgroundManager {
         toolCalls: 0,
         lastUpdate: new Date(),
       },
+      parentModel: input.parentModel,
     }
 
     this.tasks.set(task.id, task)
@@ -322,10 +323,16 @@ export class BackgroundManager {
         const messageDir = getMessageDir(task.parentSessionID)
         const prevMessage = messageDir ? findNearestMessageWithFields(messageDir) : null
 
+        const modelContext = task.parentModel ?? prevMessage?.model
+        const modelField = modelContext?.providerID && modelContext?.modelID
+          ? { providerID: modelContext.providerID, modelID: modelContext.modelID }
+          : undefined
+
         await this.client.session.prompt({
           path: { id: task.parentSessionID },
           body: {
             agent: prevMessage?.agent,
+            model: modelField,
             parts: [{ type: "text", text: message }],
           },
           query: { directory: this.directory },