I think that the provided answers give solid advice worth considering. However, since this is a vacation to Florida, I feel that someone might as well throw in some "Pros" for going with the oversized, luxury SUV.
To be clear, my hunch is that the mini-van class will probably suit your families needs, but if the rental car dealership has enough cars available you might be able to reserve a vehicle and ask if you can peruse their lot for an upgrade. If they have the stock available, usually they are willing to let you since it equates to more money in their pockets. Assuming you get that chance, here are some reasons why you might enjoy the SUV.
1. These things can be HUGE!
You've already mentioned that you are coming from Europe. I've been to Europe, from the US, multiple times on business. One of the first things I noticed when I left the airport on my first few trips was that American cars and streets are BIG, compared to European cars and streets.
I'm not complaining but it was a stark observation. Europe has had people populating and growing in that region for thousands of years. In the US, we haven't experienced a population "explosion" until a few hundred years ago. This means that America is quite spacious and our roads are wide and parking lots tend to be large to make room for our oversized vehicles.
Many people may feel that an SUV is "to much" and a ludicrous display of nonsensical, gas guzzling American consumerism. They might be right but if this is your only chance to drive something that feels like a tank, you might actually enjoy the experience! Florida is a rather spacious state, even for America. It has a lot of farmland, which means that their is plenty of room for growth around it's cities. As such, you shouldn't have any trouble learning to use an SUV, versus a small, family sized vehicle because the roads and parking lots can easily fit these larger, common vehicles.
2. SUVs are often manufactured to be semi-luxurious, yet family-friendly vehicles.
The SUV comes in all shapes and sizes but they are often equipped with bigger, cushier seats and better sounds systems. They are more likely to have drop-down TV consoles for your kids to watch TV (very handy when you are on long trips, with little ones!) and the A/C might be a little better since the vehicle has a more powerful drive train.
Florida can get hot in the late-spring to early-fall months. In Florida, there won't be a car on a rental lot without A/C but the oversized SUV might cool down quicker than some of the smaller options due to the extra power under the hood. Which brings me to point #3.
3. More power!
The SUV was originally designed to be a hybrid passenger vehicle + work truck. Oversized engines were put in them so they could pull considerable weight but also function as a day-to-day passenger vehicle when not towing.
Some time around the late 90's, manufacturers realized that that if they made these vehicles a little more consumer friendly they could market this vehicle to those who like a combination of power+luxury. This is when the Escalades, Excursions and Hummers started to come out as consumer vehicles. The population picked up on it and a new type of consumer vehicle was born.
Part of the "luxury" feel is having a big vehicle that can get-up-and-move. The hummer was initially designed for the US Government for military use across aggressive terrain. When the GMC company began making consumer-grade, luxury versions of the vehicle they left the power with the vehicle (and it shows in the gas mileage!) and all other manufacturers of luxury SUVs followed this trend.
This could be a turn-off if you are environmentally conscientious because some of these vehicles will get as little as 10 miles per gallon (23.5 L/100 km) under worse conditions. That's worth considering. However, if you have the extra funds for the fuel and you think you might enjoy the extra power from the pedal, the SUV could make for a fun driving experience.
The oversized, luxury SUV might be too much for you and your family. There are smaller sized SUVs and, from my experience, many rental companies tend to have Japanese vehicles on site which also tend to be slightly smaller, and more efficient, than their American contemporaries.
A mini-van will certainly meet your needs but an Oversized Luxury SUV will add to the experience of your vacation if you are willing to try to drive the larger vehicle. If you take a look at one on a rental lot and you think you'd rather not try to drive a vehicle the size of a european passenger bus, you might want to get a smaller sized SUV.
Most of these smaller vehicles have some of the luxury options, but not all of them. The smaller size shouldn't be an issue for you because they should seem rather big, considering you are likely more familiar with European sized, smaller vehicles.
I wouldn't worry to much about navigating the bigger vehicles in Florida. It is a tourist-friendly state. Many of the places you will want to visit understand that people get in their BIG vehicles and drive down from regions as far as Canada to come and visit. Because these tourists come from so far away, they often come with oversized vehicles to haul all of their road gear.
I hope you enjoy your vacation. Regardless of the vehicle you pick, you should have many transportation options and I'm sure you will have fun, even navigating our American highways.