Art. 135 of the Italian Vehicle Code specifically states that the fine for Driving without an IDP or official translation goes from a minimum of 400€ to a maximum of 1600€.
It is impossible that a fine not paid within 5 days increases fivefold (from 280€ to 1400€). The law is very clear on this point. You have 60 days to pay the fine without any further penalties.
What is true is that you get a 30% discount on the fine amount, if you pay it within 5 days, instead of waiting for the full 60 days.
So, assuming the story from the OP is real, he got a 400€ fine for not having an IDP (the minimum by law) which is reduced by 30%, or 120€ less (400x30%=120) if paid within 5 days.
400€ - 120€ discount = 280€ (Reduced amount if paid within 5 days).
If not paid within 5 days, then the discount does not apply and the full amount of 400€ (not 1,400) is due within 60 days.
If not paid or contested within 60days, after the 61st day, the fine increases by 50% of the maximum provided by law. Since the maximum here is 1600€, then the unpaid fine would be 1200€ (400+1600/2=1200).
So, yes, the $25 fee from AAA is cheap in comparison and I don’t know why rental car companies do not make it know when a specific Country or even US State requires it (in the US some states require an IDP for foreign drivers, others do not).