Fix register-docs feature not being tested

[Bromeon]

The doc registration from RustDoc feature was added with an unconventional test setup, which has since posed a multitude of problems:

• It was a unit, not integration test (checking whether an XML file was written with expected contents).
• It worked without Godot at the time, but since then, ClassName uses StringName internally and is no longer usable in unit tests. I moved it to integration tests.
• The test relied on only one specific class to be registered, calling next() on the plugin iterator. I changed this to find the correct class instead.
• It being feature-gated behind register-docs meant that regular cargo test wouldn't run it, and neither CI nor check.sh were updated to include that feature. So it has effectively been without tests through the different additions.
• Since it hasn't been tested, a change in #918 has slipped.
• Clippy hasn't run on it either, so I patched several violations.
• The feature was inconsistently called docs and register-docs depending on crate, adding complexity.

[GodotRust]

API docs are being generated and will be shortly available at: https://godot-rust.github.io/docs/gdext/pr-942