Eight months after the post, but for what it's worth...
There is a vague implication that the school does need to be in the US.
An eligible elementary or secondary school is any
public, private, or religious school that provides elementary or
secondary education (kindergarten through grade 12), as determined under
Now, understand that Coverdell and 529 are "loosely regulated" as far as the IRS goes. It's pretty much honor system. The IRS does not have the resources to enforce all the rules for every taxpayer who utilizes them. I have never seen anything ever written where the IRS came down on someone for improperly applying funds when it came to tuition. Particularly when it comes to Coverdell, in that it has been around longer and has always been broader in what expenses were qualified. Do they look for where the payment went? Verify that the amount withdrawn is correct for the qualified expenses? No, and no.
I am not a tax professional, and I am not giving any advice here. However, if it were me, I would go by "Don't ask, don't tell". Likely the only way it would possibly come up is if you were audited for some other reason and they did an in-depth audit (as opposed to one where they simply tell you some mistake(s) they found and how much you owe). Both unlikely for most folks. There is no reason that the Coverdell withdrawal would raise any flags on its own - so nobody is going to be checking that payment went to a US-based school.
Have the Coverdell issue payment to you for the withdrawal (better to have them mail a physical check made out to you), then you make the payment to the school. Keep copies of both the withdrawal (account statement) and the payment (check copy or credit card statement) just in case in the unlikely event there are questions down the road.