This is not only a problem for freelancers. In Germany, all the one-man businesses and partnerships (GbR in German, VOF in Dutch) and all 'Gewerbes', (labour focussed companies, like bakers, painters, plumbers, dentists, shop-owners, builders, architects and many many more) operate without the chamber of commerce registration number. They do have to pay a fee, though, but they don't get an entry in the register.
There are thousands and thousands of companies in Germany that don't have this Handelskammernummer. And applying for such a number is way too much effort, just for a change of banks. It means transforming to another business entity.
Can you please consider identifying companies by their VAT Registration Number? Most companies already have such a number - and if not, they can apply for one for free. VAT Registration Numbers (Umsatzsteuer Identnummer, UStIdNr) are unique and can be verified for free also.
Here you can check if the number exists: https://evatr.bff-online.de/eVatR
If the number does exist - you can check if the address people give you, belongs to that company. There is an interface to do this automatically.
Hope this helps!
Can't wait to kick out the old-fashioned banks and shift our company to Bunq!