fix: use specified HTTP codes during exceptions on public/remote endpoints

[joshtrichards]

• Resolves: #56052

Summary

We're already specifying the HTTP codes, but have been ignoring them.

TODO

• [ ] ...

Checklist

• Code is properly formatted
• Sign-off message is added to all commits
• [ ] Tests (unit, integration, api and/or acceptance) are included
• [ ] Screenshots before/after for front-end changes
• [ ] Documentation (manuals or wiki) has been updated or is not required
• [ ] Backports requested where applicable (ex: critical bugfixes)
• [ ] Labels added where applicable (ex: bug/enhancement, 3. to review, feature component)
• [ ] Milestone added for target branch/version (ex: 32.x for stable32)