Cash

I think that you will find that the amount you get "dinged" for International charges is far less than what you lose at an exchange house, but if you can, check their rates.

Also, the International charges vary greatly between financial institutions. Banks like Chase, USBank, and some other can be as high as 3%. A lot of credit unions just pass on the 1% charge from what's called "the network". Well Fargo (for customers) charges $5 for a withdrawal of up to $500, so about 1%. A few brokerages eat the 1% charge as a service to you.

These charges were in effect the last time I checked.

You're paying what, maybe 3-4% (maybe more?) extra buying your Euros ahead of time? Just so you can avoid paying maybe half that in Europe? And you're planning on carrying $2500 in Euros around with you? And then you'll pay 3-4% converting any leftover Euros back to dollars? I don't think it makes any sense financially, but aside from the security concern of carrying so much cash around, you're just out maybe an extra $100 doing it your way.

There are a lot of searchable threads on ways to save. There are debit cards like Schwab that cover your international fees.

I can't really tell you what your daily budget should be. I don't know how you travel, your destinations, your season or the number of people. There's a thread here about it. https://community.ricksteves.com/travel-forum/italy/daily-budget-for-italy

A credit card is a really useful travel tool. Your debit card doesn't offer nearly the protection that a credit card would.

A cel phone with data is also really useful.

Perhaps open a new bank account with just enough in it to cover your trip and have it come with a credit card.
I would not bring that much cash with me….what if it is all lost, then what?
If you do bring it, be sure to split it up among your group so that no one person is carrying it all.
And….bring more than one debit card , for two different accounts.
If you lose a debit card it’s a much bigger problem than losing a credit card.

I’m curious how you manage at home with no credit card?
Also, if you have an emergency in Europe, need to rebook flights, pay for something like car rental or any number of scenarios, you are going to need a credit card.

Do let us know what you decide in the end.

I think that you will greatly benefit from embracing the XXI century and start using debit/credit cards to pay at least the more significant expenses, like lodgings and restaurants. I do use hard cash for smaller purchases, like an espresso or cappuccino at a coffee bar or for the occasional gelato, but exchanging dollars for hard cash is costly. Besides the exchange rates applied you will likely have to pay a withdrawal fee each time you use your ATM card to withdraw money. I usually withdraw cash from my Italian bank but if I used my Chase bank they would charge me $5 per withdrawal.
You don’t need to use credit cards, but a debit/ATM card would be useful. The 3% fee is probably less than the fee you will pay to exchange hard cash.

Honestly, so much is paid for with a credit card in Europe these days, that hardly anyone carries much cash. Is there a reason that you cannot shop around and get a credit or debit card with little or no international transaction fees? It would make everything so much easier.

Since your hotels and train tickets and excursions are already paid for, what do you anticipate spending cash on? Even with four people, that $2500 is $625 each, so are you thinking that this would be for food?

I think there is some confusion. The exhange house charges a $5 fee for the transaction, then whatever is the current rate of that day. For example $100 is worth maybe 87 euros, so my cash is not $2500 but 2125 euros. I OWN a credit card but opt not to use one, but have one for an emergency. That sum divided by 13 days amounts to 167 euros per day for 4 people that will need to eat and pay for other essentials. I was actually wondering if Italy is trending towards cashless, recently at a Yankee game and a day trip to Lego land no cash was allowed.