Support custom payload fields in the agentcore dev Web UI

### Description

Thanks for building AgentCore — it’s been really helpful for developing and running AI agents.

The `agentcore dev` Web UI currently seems to support only a session ID and prompt.

It would be useful to also provide custom request context. For example:

```json
{
"prompt": "Why did this change?",
"context": {
"current_menu": "sales_forecast",
"reference_date": "2026-08-01"
}
}
```

In real applications, agents often need request-specific context such as the current page/menu, reference date, selected item, or filters. This context can change between requests, so a session ID alone is not enough.

Currently, this can be tested by calling `/invocations` directly, but not through the dev Web UI.

Would it be possible to add an optional JSON payload/context editor to the `agentcore dev` Web UI?

Thanks again for all the work on AgentCore!

### Acceptance Criteria

- [ ] The agentcore dev Web UI allows users to provide additional custom JSON fields alongside the prompt.

### Additional Context

_No response_

Start by tracing the `agentcore dev` Web UI request flow and compare it with the `/invocations` payload described in the issue. Add an optional JSON context editor alongside the prompt, and verify that submitted custom fields reach the agent while existing session ID and prompt behavior still works.

Written by the indexing model from the issue text.