[badge_reader] Add lock to loginState.getProfileType() API

For extensions installed on the lock screen profile, return the correct
profile type instead of the default USER_PROFILE - this is needed for
Badge Based Authentication lock/unlock functionality.

Bug: 441909493
Change-Id: I10226d21c9c7573dc6b5aeb42068ed48a7677a27
AX-Relnotes: Adds a new enum value for getProfileType() API.
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6958870
Reviewed-by: Marc Treib <treib@chromium.org>
Reviewed-by: Devlin Cronin <rdevlin.cronin@chromium.org>
Commit-Queue: Aida Zolić <aidazolic@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1526150}
NOKEYCHECK=True
GitOrigin-RevId: 66e2f0a78d848609bcc09d6f8c74f2e4433602c4