Fix refill unblinded tokens after captcha solving. (uplift to 1.53.x)

[aseren]

Uplift of #19027
Resolves brave/brave-browser#31182

Pre-approval checklist:

• You have tested your change on Nightly.
• This contains text which needs to be translated.
  • There are more than 7 days before the release.
  • I've notified folks in #l10n on Slack that translations are needed.
• The PR milestones match the branch they are landing to.

Pre-merge checklist:

• You have checked CI and the builds, lint, and tests all pass or are not related to your PR.

Post-merge checklist:

• The associated issue milestone is set to the smallest version that the changes is landed on.

[tmancey]

LGTM++

[kjozwiak]

As discussed on Slack between @btlechowski & @tmancey, we'll let this one ride the trains in 1.55.x via #19027. If we get enough user reports that's due to the above, we'll uplift it as needed.