Yeah I don't really understand the part about not being able to invoice either. IANAL but it's the other way around - you need a VAT number to invoice clients in Germany (and the EU), not outside of it. VAT is exempt for clients outside of the EEA.

Also: I've always used a ZZP structure (one man company - Dutch version) for mine, not a BV (LLC), because there's a thing called Professional Liability Insurance. But maybe it's different in Germany? I can't imagine that doesn't exist there though.