Others can work you through the prices, but I can help you understand the timing issues on your journey from Rome to Lauterbrunnen.
I'm willing to bet that the train will be easier, and may well be cheaper, than the flight you have outlined. Remember that the train has no security theatre to put up with. You just get on (after ticketing and any needed validation).
The 8am train from Roma Termini (ES35610) is a non stop, FrecciaRossa 300 kph train which gets to Milano Centrale by 10:55 which gives you an almost perfect amount of time (28 minutes) to get to the Swiss EuroCity train (EC52) which gets to Spiez in only 6 stops at 13:53, with a 12 minute connection at Spiez to the local train to Interlaken Ost, arr at 14:28. No sweat if you miss the connection at Spiez, the trains are frequent and easy to get.
Anyway, that's 6 hours 28 minutes, city centre to city centre. From Interlaken Ost (don't get off at the first Interlaken station of Interlaken West, that's all the way across town from the little trains up the mountains) the route is identical.
Can you beat that by flying? It will be close.
So - same starting point, Roma Termini.
Every half hour(at xx:05 and xx:35) is the Leonardo Express (Regionale but expensive at €14 per adult) which takes 32 minutes to the airport, and you need walking time and at least 2 hours for an Intra-European flight. So shall we say 3 hours from Termini to departure? Then an hour for the flight, 30 minutes to get out of the airport and on the bus and half an hour on the bus (rounding up) to Basel SBB Hbf. Trains from there take 2 or 3 minutes short of 2 hours straight through to Interlaken Ost and go every half hour at xx:59 and xx:31. Add half an hour in case the flight is late or the bus is just missed, or luggage takes a long time.
So, 3 hours plus one hour plus one hour plus 2 hours is 7 hours, or half an hour SLOWER than the train.
Granted, I have made some allowances on the flying option, but having flown around Europe enough I think they are fair. I also picked the fastest nonstop Freccia that there is, and if others have 3 or 4 station stops it will add a few minutes, but the result will be similar. You can also plan to add time in Milano to have a meal, but I'd just get nibbles for the train. And you can add another hour or so to the train by going through Luzern and via the most fabulous narrow gauge cogwheel train from Luzern to Interlaken via the Brunig Pass - the kids and you might love that train as much as I do. But you can always do that on your way out for your planned time in Luzern.
I can't tell you if the kids would prefer Rome or Lauterbrunnen Valley or Luzern. There are a million kid friendly things to do at each.
You know your kids. What do they each like to do? Bike ride down a mountain? Climb on a glacier? Ride a revolving cable car? Ride a steam boat on a lake where you can actually see the steam engine driving the paddle wheels and watch the different bits move? Be a gladiator? See the Pope or his house? Walk in the 2000 year old cart tracks on an ancient road with its original "paving"? See where Caesar was assassinated? See the 75 cats which live there now? Walk across an ancient painted bridge? Go to one of the top transportation museums in the world which is really hands on?
Is one (or both) of the kids a real girlie girl or a real sporty type, or some other broad description?
Tell us and we can help....