When dealing with a large data volume (e.g., 100,000 child records per parent record), triggers are not an ideal solution because they can quickly run into governor limits such as:

SOQL Query Limit: Triggers have a limit on the number of SOQL queries that can be executed per transaction.
DML Statement Limit: Large DML operations can exceed transaction limits.
Using a batch job is the recommended approach because it allows processing large datasets in manageable chunks, avoiding governor limits. Additionally, by scheduling the batch job using Database.Schedulable, the update process can be executed asynchronously.

Why not use fomular field?

The answer is:

B - A Database.Schedulable and Database.Batchable class that queries the Catalog object and updates the Catalog Items if the Catalog CurrencylsoCode is different

C - A Database.Schedulable and Database.Batchable class that queries the Catalog Item object and updates the Catalog Items if the Catalog CurrencylsoCode is different is not a correct option because it is less efficient and more complex than querying the Catalog object. If you query the Catalog Item object, you will have to retrieve all the Catalog Items in your org, which can be a very large number, and then compare their CurrencylsoCodes with their parent Catalog’s CurrencylsoCodes. This can result in more SOQL queries, more CPU time, and more memory consumption. On the other hand, if you query the Catalog object, you will only have to retrieve the Catalogs that have changed their CurrencylsoCodes, which can be a much smaller number, and then update their related Catalog Items. This can result in fewer SOQL queries, less CPU time, and less memory consumption. Therefore, querying the Catalog object is a better option than querying the Catalog Item object....