In your case, the simplest solution is indeed to have your visitor handle the paperwork under their own name as a non-EU resident. Depending on the store, make sure they have their passport with them, and possibly proof of residence abroad.
As far as I know, the name on the credit card used for payment doesn’t really matter—unless the store has a specific policy requiring the cardholder’s name to match the buyer’s. But that’s up to each store.
The tax refund itself can be issued to any account or card: I personally live outside the EU but keep bank accounts in the EU, and my refunds often get sent there. In the past, I’ve also had family buy things as gifts, and we had the tax-free forms filled under my name. We then validated them at customs the next time we traveled out of the EU (just remember you must do this within 3 months).
So yes—you could pay, have the paperwork filled under your visitor’s name, and they’d get the tax refund as long as everything is done properly.