Bug 1862955 - fix browserstack session tests

[rosahbruno]

This is a followup to #1850. The changes in that PR broke the browserstack tests. This updates our browserstack tests to rely on data from LocalStorage rather than trying to pull session IDs from log messages.

Pull Request checklist

• Quality: Make sure this PR builds and runs cleanly.
  • Inside the glean/ folder, run:
    • npm run test Runs all tests
    • npm run lint Runs all linters
• Tests: This PR includes thorough tests or an explanation of why it does not
• Changelog: This PR includes a changelog entry to CHANGELOG.md or an explanation of why it does not need one
• Documentation: This PR includes documentation changes, an explanation of why it does not need that or a follow-up bug has been filed to do that work
• Documentation should be added to The Glean Book when making changes to Glean's user facing API
  • When changes to the Glean Book are necessary, link to the corresponding PR on the mozilla/glean repository
• Documentation should be added to the Glean.js developer documentation when making internal code changes