What you might consider is this. You don't have to camp in the National parks. There will be lots of private campgrounds and smaller state campgrounds in fairly close proximity to the big national parks. You can stay at one of those then day trip in. That is how we do it, we very rarely make reservations.
With a 22ft, if there is any site left in just about any campground you will fit as opposed to a much larger rig.
4th could be tough regardless, that might be the one date or dates you might consider reserving. Good luck and have fun
PS if it were us I would book your first night somewhere, and your last nights over that 4th time
One last thought I have never driven in Italy, so I don't know. But I have driven all over the UK. I thought RV parks were pretty few and far between. Not so in the US they tend to be pretty much everywhere.
When we camp with the grand kid, we pretty much stay at KOA's for a number of reasons.
1) they are kid friendly with pools etc
2) quality is always at least OK
3) KOA' have an APP you can get. So what we will do, is in the morning, figure out, what we want to see that day and where we want to end up. We find a KOA in close proximity and make an online reservation. Takes the worry out of where are we going to stay that n night. And of course your least chance of finding a campsite is going to be showing up late, so making a reservation in the AM, generally there are always sites available