Add Sentry Client HTTP PUT capability

What does this MR do?

This refactors the way we call the HTTP client within the Sentry::Client class. It DRYs up the code by allowing us to not redefine the error handling for each HTTP method.

We've also added in the ability to do a PUT. This is needed for #39825 (closed) and #39665 (closed)

The existing tests for the HTTP GET are still valid, while the HTTP PUT will be tested with MRs from the above issues.

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

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