[beman-tidy] Add CI tests which actually build beman-tidy and then use it

[neatudarius]

[beman-tidy] Add CI tests which actually build beman-tidy and then use it to prevent issues such as https://github.com/bemanproject/infra/issues/144

Probably here https://github.com/bemanproject/infra/blob/main/.github/workflows/beman-tidy.yml#L36, after Run tests, we should also add Build and install.

[neatudarius]

Tests

• On-demand run (with broken tests) - https://github.com/bemanproject/infra/actions/runs/16438603107

• On schedule run (with broken tests) - https://github.com/bemanproject/infra/actions/runs/16436594640

• On-deman run (with passed tests) - https://github.com/bemanproject/infra/actions/runs/16438678422

• On schedule run (with passed tests) - waiting tomorrow's scheduled task

• On push - https://github.com/bemanproject/infra/actions/runs/16438821791

and now the badge is green