Child sleeps in same bed with us - what kind of room to book?

I would make sure those familiar with Italy chime in here.. Ive been but only twice.. however in Paris , even with a child in your bed you MUST book a triple or risk being refused the room on arrival..It happened to a family that arrived with an 18 month old.. because there are fire occupancy limits for each room , and since hotel had no more triples left they had to send the family away.

Now.. that does not mean they will always charge you as a triple.. you just have to make sure they will allow all three in room.

Since you are on a budget I think you will have to settle for two twins pushed together.. and they may not be sheeted together.. Queen beds in budget hotels very rare.

Just an aside.. I realize dd is used to sleeping with you at home.. but she might be persuaded to try sleeping in her own bed if its pushed right beside yours because shes in a different place.. sometimes a different place has different rules.. like you can walk barefoot at home but not at school for instance. Good luck.

In Europe, you need to "declare" all your room occupants, regardless of age. So let them know you're a party of 3 and you'll be sleeping in 2 beds and see what they say.

Yes, unlike in much of the US, a child counts in the occupancy rate for a room. Often a double (two twin beds) will not allow 3 people, even if one of those people is 3 years old. Some will allow children in doubles for a small nightly fee. Be prepared, though - you may need to book a triple at some places. It will take some looking around to find a double that will allow 3 people. I had some luck getting my son allowed in doubles when he was under 2-2.5 by requesting a crib. He slept in the bed with me anyway, but a crib signified baby and many places were less likely to charge for a baby. This issue is actually one of the more difficult one for travelers with children - after child 2 was born we pretty much stick to apartments because quad rooms are smaller, harder to find, and often more expensive.

Another option would be to rent an apartment on air BnB or VRBO. Maybe you find one not more expensive than a hotel meeting your needs.
Besides this, it maybe the occasion for your daughter to try something new in another framework.

If you check a booking site, like booking.com, the occupancy icon will show how many people the room can accommodate, filter by two adults plus one child. The site also asks for bed preference, so you can request a queen bed if the hotel has them.

Unfortunately if using online sites you do need to declare a party of 3 which will mean booking triple rooms even if you won't use the third bed
Keep in mind breakfast is also included for most hotels in italy so that is also an expense they incur per person
If you know the hotel you want to book you have more options
Look at the available rooms on their website and send an inquiry asking if you can book a double room for your party of 3 if you share existing bedding

I have done this multiple times and they were fine with it
Some had an extra fee, others did not
Some small rooms they may say are too small for 3 like your basic double rooms but if you go with a superior double it will be ok
In some cases the cost of a much nicer suite with a double bed and a couch may be the same as a standard triple
I did this is one hotel where I saw a really nice suite (frescoed ceilings and the like that were unique to this one room) that was listed as max occupancy of 2 on their website
They emailed back in a couple of orders offering I could book that room with existing bedding at no extra charge or for 20 euro per night they could add a cot to that room for us since the room was large enough
Main point is to notify the hotel directly in advance and you will be fine and they will let you know what is possible

Another tip about booking.com. When you put in the number of adults and children, the website will ask about the age of the child. When you put that in it will help to filter the results to better suit your family.