Write operations (fill/click) fail with "No tab with given id" in Helium, but work in Chromium

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I tested this on two browsers and the results are: it works in Chromium, it doesn't work in Helium (both Chromium-based, Browser MCP extension v0.1.3 + server @browsermcp/mcp@0.1.3, macOS, accessed via npx opencode-browser).
Repro steps:

1. Connect the Browser MCP extension to a tab and run navigate
2. Run snapshot — works fine
3. Run click or type (fill) on an element from the snapshot — fails
   Actual behavior:

• In Chromium: all of the above work.
• In Helium: navigation and snapshotting work, but click and fill/type just return errors:
  Error: No tab with given id 1995387775
  (the id stays the same stale value no matter of which tab is connected)
  Expected behavior:
  The fill and click tools should work in Helium exactly as they do in Chromium, since both are Chromium-based.