Should I ever accept "pay in dollars" at an ATM or card machine?

Never. Always choose euros. That offer is dynamic currency conversion: the machine or terminal converts for you at a rate it sets, typically 3–7% worse than your bank's, and pockets the difference. The screen presents it as a helpful certainty — "you'll be charged $214.30" — which is exactly why it works. Decline, choose to be charged in euros, and let your own bank do the conversion. The same trick appears on restaurant card terminals and hotel bills. It's the single easiest way to lose money in Italy without noticing.