Follow-up from "Fix project access token regression"

What does this MR do?

I created !47247 (merged) to fix problems with project access token HTTP authentication (issue: #259665 (closed)), but that introduced more problems with project bots pulling containers (see #296028 (comment 477726552) for more details).

This MR fixes the problems introduced by !47247 (merged), as well as the HTTP problems that !47247 (merged) was intended to fix.

Screenshots (strongly suggested)

Does this MR meet the acceptance criteria?

Conformity

• Changelog entry
• Documentation (if required)
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Availability and Testing

• Review and add/update tests for this feature/bug. Consider all test levels. See the Test Planning Process.
• Tested in all supported browsers
• Informed Infrastructure department of a default or new setting change, if applicable per definition of done

Security

If this MR contains changes to processing or storing of credentials or tokens, authorization and authentication methods and other items described in the security review guidelines:

• Label as security and @ mention @gitlab-com/gl-security/appsec
• The MR includes necessary changes to maintain consistency between UI, API, email, or other methods
• Security reports checked/validated by a reviewer from the AppSec team

Related to #296563 (closed)