Regional vs National trains with kids/pricing - I'm dense

A Regional train is called Regionale or Regionale Veloce and the only fare on such trains is called Ordinaria.

I doubt you have made a search on trenitalia.com for those routes and picked the slowest, non direct trains.

Will they let me on

Nobody's checking your tickets before getting on. A conductor may or may not come at your seat during the journey and will ask you to show the PNR code and the kids' passports. If one of the kids is older than 15 and you have a discounted tickets for kids younger than 15 you'll be fined. You can hand your passport and refuse to pay on the spot.

which I don't think was a choice when I did the other two

Discounted tickets are capacity controlled and may sell out on some routes.

Thank you Dario.
I booked faster trains.

Now I look at the site as well - that says "Children under the age of 4 travel for free in both first and second class, but are not eligible to their own seat. Children under the age of 15 (the maximum age limit is counted from the day after the child’s 15th birthday), may occupy their own seat, and can travel with a 50% discount, or 30% off sleeping cars (VL) and couchettes (CC) services. In addition, you can take advantage of the Bimbi Gratis (Children Travel for Free) offer, dedicated to family groups of 2 to 5 people, which allows children under 15 to travel for free. For more information, please visit the page dedicated to the Bimbi Gratis offer."

Both the ones that I have booked are "Bimbi Gratis"
So the kids will be 12, 14 at time of travel. Sounds like we should be okay then?

Don’t overthink this. If you listed your kids ages correctly when booking the system will have sold you the right tickets.