I am in the US and have been using Fastmail for years. I have no problem with automated billing to a consumer credit card. I have recently begun using Hetzner as well, and this also bills to the same card. I just looked at the Hetzner charge, and it shows up in the CC transactions as "hetzner.com" with a US domestic phone number associated.
My guess is that your CC has more strict foreign alerting than mine, so perhaps this is not very helpful information. But it is certainly possible, and perhaps Hetzner billing support (and Fastmail too!) have some advice for how to smooth things over with the bank.
The non US companies I have tried this with in the past also had a US number, but their receiving account was non-US and I am fairly certain that is what they trigger off of. The protection in place is intentional and if anything I have asked the bank for even tighter controls. I want something closer to the 3rd parties that allow making specific rules, unique card numbers for specific vendors but getting a bank to do that is difficult. My card does not have strict foreign alerting, it has explicit blocks of non US that I put in place. If someone outside of the US tries to use the card, it completely disables the card until I walk into the bank which is not a problem as they are just down the road and they all know me. The only option with Fastmail is to disable my protections for a day. I don't mind that as they have earned some limited trust from me and they are inexpensive. VPS and server providers can get more expensive really fast so I am less inclined to do that unless they show up as a US entity as far as the bank can see. That's why I was asking if others here also had to disable their protection or if they know for certain they show up as a US entity.
My guess is that your CC has more strict foreign alerting than mine, so perhaps this is not very helpful information. But it is certainly possible, and perhaps Hetzner billing support (and Fastmail too!) have some advice for how to smooth things over with the bank.