I've got a invoices table where the company name does not actually appear as a standalone field - the numeric primary key for the company is there, but not the name. The data is pulled over via the relationship.

If I want to use fmSearchResults for searching in the invoices table can I select the related table for one of the search Text Fields? Actually, I know I can, the question is should I?

The "should" is just a matter of speed. You certainly can, and it will be a slower search than looking for an indexed field in the invoices table, but depending on your setup and the number of records you have it might be OK.

FWIW, many people lookup things like this into the invoices table so the invoice is preserved as-written in case you delete or edit the customer record.