Is there a system that can cope with 3 different businesses run by the same person and lots of bank accounts
Absolutely. It depends on what industry you are in but Blue Link ERP is a system that handles a virtually unlimited number of companies and the first 5 companies for the same price as 1. Many bank accounts are not an issue:
Yes, Passport Accounting software will accommodate virtually unlimited companies and unlimited bank accounts. Allows printing checks on blank check stock. No wrong or missing checks, this works great for property managers.