I'm confident the property charged in Euros. I'm having some difficulty figuring out if it's an exchange rate issue as all I can see (on both my US card issuer's app and website) is the transaction in dollars. When I called my credit card issuer, I was told the amount in Euros and the rate would show up on my statement, but they couldn't tell me that information at this time. When I apply the exchange rate on the date of the transaction, the dollar amount of my charge roughly corresponded to the 820€ I should have been charged. Ignoring exchange rates for a minute, I don't understand why the property would only charge 795€ when the rate I agreed to, 820€, was 100% prepaid and non-refundable.
If your bank told you the amount charged in euros was 820 euros, then the hotel got 820 euros and the issue isn't related to exchange rates.