This doesn't matter. For B2B, if you don't show your EU customers a valid invoice, they simply cannot pay you.
It's not EU governments or post-purchase issues you have to deal, most of the time - it's trying to persuade a large accounting department to pay you unaccountable money. If you can't follow their invoicing requirements, it's just not going to happen.
Do people ever stop and wonder WHY in nations like Portugal the informal economy is roughly DOUBLE of say the US? When every invoice must be available to the government, what actually ends up happening is non-conforming invoices either get made up by the customer or the money magically flows out under some other auspice.
Making it literally illegal to accept an invoice that is legal in the country in which you obtained it borders on logic even my toddler can understand is absolutely begging for bad business climate or tax evasion.
What is the mechanism for French government to subpoena a non-EU business to find this invoice? If the customer (instead of vendor) produced an invoice that matched their bank statements and non-conforming receipt, hypothetically, would anyone really know the difference?
Probably they wouldn't as full audits aren't that common, but audits do happen and most accountants won't care about your purchase as much to personally commit a felony and forge documents for no good reason (accountants do have personal responsibility, and every accounting course reminds wannabe accountants that "the boss ordered it" is not an excuse) so they simply say that they can't/won't do it and if the vendor can't send a satisfactory invoice then they won't do business with that vendor.
How would a French audit uncover anything wrong with a conforming format invoice under the name of some random American company that exactly matched the bank statement, physical goods, and customs paperwork?
I don't think US has many relevant freedoms left. Some people cling very hard to legacy 18th century ones while the Moloch just gobbles up all the new ones stemming from progress.
It's not EU governments or post-purchase issues you have to deal, most of the time - it's trying to persuade a large accounting department to pay you unaccountable money. If you can't follow their invoicing requirements, it's just not going to happen.