rustdoc: Add gui-test for "copy item path" button

[aDotInTheVoid]

Ideally, we should have a test for this in tests/rustdoc-gui, but it doesn't seem like there's a way to inspect the clipboard. CC @GuillaumeGomez

I don't think it's worth blocking this fix on having a test, but we should add one.

Originally posted by @aDotInTheVoid in #137068 (comment)

#137048 was fixed in #137068 but without a regression test. This issue it to track adding that test.