Legal API: add business search validation to firm registration filing

#### Description

As part of a firm registration filing, the UI may perform a business search to fetch some business data.

When firm registration filings were implemented, the Legal API did not run the business search itself to validate that the data provided by the UI is correct.

This ticket to add that search to the Legal API validation for firm registration filings.

See also #21104.

#### Impact

A nefarious actor (with API access) could post a registration with invalid business data.

Trace the Legal API validation path for firm registration filings and compare it with the business-search behavior described in #21104. Verify how the API receives and validates business data, then confirm that invalid data is rejected; the work is done when registration validation performs the search and prevents mismatched data.

Written by the indexing model from the issue text.