For a non-U.S. resident with a U.S. LLC, is it better to open a business bank account in the U.S. or Europe?

Yes, a personal visit is required. You will most likely never be declined. Opening a bank account for a Non-US Citizen is very simple. A copy of your passport (with a valid Visa), any other form of secondary ID (credit card, driver's license, etc. this is optional) and some money to deposit the funds. That's it. The KYC paperwork is very simple and easy. You would be pleasantly surprised how easy and smooth the whole experience would be.

For an LLC/Inc, you are required to provide the necessary company formation documents, the seal, your EIN and Certificate of Incorporation as given by the State. All members enlisted in the LLC/Inc usually need to be present with their IDs.