What you re experiencing is a symptom of larger underlying issue.
It appears that you have happened upon a difference in Company Name that is causing you grief in a business sense --Who belongs to this invoice.. If you are not sure of which Customer is which, then you have a major issue. A viable business who sells/invoices and is paid by Customers will not remain viable indefinitely, if it is uncertain of whom to invoice.
The whole idea of your database is to support your business. And in this case it is to identify and invoice companies for services/products. You have to get unique identifiers on your Customer records, and use/maintain them accurately and consistently in all of your processes.
You and/or your colleagues/bosses/designers will have to identify WHAT exactly idenfies a Customer; then adjust your database/table(s) etc accordingly.
I have worked in orgs where they merged multiple, disparate systems involving companies, producer, fabricators, vendors, suppliers.... The approach was to look for common strings in name(legal, operating, alias), address street, city, postal code/zip, phone number.. The idea was to apply a weighting factor to each of the terms of comparison. If the final weight was above X, then these were accepted as same with x % confidence. If the results were less than X, they were considered not the same. In many cases companies were phoned to confirm information.
Once you get the unique identifiers assigned, it is critical to keep them current.
I don't know what steps will apply to your particular set up, but I wanted to highlight why getting customers uniquely identified in your business and operational databases was important.
Good luck with whatever strategy you choose.