Make the existing SQLite fallback reliable and testable when Docker is unavailable

## Description

The functional test infrastructure already contains a SQLite fallback when the SQL Server Testcontainer cannot be started.

However, the current behavior is not fully reliable or consistently testable:

- `CustomWebApplicationFactory` falls back to SQLite only for a specific exception type.
- `DockerAvailabilityTests` fails the complete test run when Docker is unavailable, even though the remaining functional tests can use SQLite.
- There is no explicit test verifying that the SQLite fallback is selected and works correctly.
- The active database provider is not clearly visible in the test output.
- The current provider selection relies partly on a global environment variable.

## Desired behavior

- SQL Server through Testcontainers remains the preferred provider for full functional test coverage.
- The functional test suite can run successfully with SQLite when Docker is unavailable or when SQLite is explicitly selected.
- The SQLite path is covered by automated tests.
- Docker-specific coverage does not make the SQLite fallback test run fail.
- The selected database provider is clearly diagnosable.

## Possible approach

- Make the test database provider explicitly selectable, for example through a factory option or dedicated test factory.
- Add a test verifying that `AppDbContext` uses SQLite in fallback mode.
- Add at least one endpoint test that runs against SQLite.
- Change the Docker availability check so that it does not fail a valid SQLite-only test run.
- Keep SQL Server/Testcontainers as the default or CI reference path.

Start with CustomWebApplicationFactory, DockerAvailabilityTests, and AppDbContext to trace provider selection and the Docker-unavailable path. Run the functional tests and add coverage for SQLite provider selection and at least one endpoint. Done means SQLite-only runs succeed, Docker-specific checks do not fail them, and the selected provider is visible in test output.

Written by the indexing model from the issue text.