So, here is the response I received from Github support:
"I’m afraid it’s not possible to pay for your sponsorship in EUROs at the moment. We use Stripe to faciltate Sponsorships and we send your payment (made in USD) to the recipient’s account in USD. This means any fees for the currency conversion from EURO to USD will be set and charged by your card provider. If the recipient of your sponsorship receives their payout in a currency other than USD, Stripe will apply “the daily mid-market rate provided by our service providers” and take “an additional fee of 2%” from your sponsorship to them. You can read more about Stripe’s currency conversion process here:
https://stripe.com/docs/currencies/conversions
I’m sorry I don’t have better news to share here, but I will highlight to our engineering team that it it would be beneficial to avoid conversions, if possible."
So, maybe in the future, Github will facilitate this.