Rome to Capri with a stop at Pompeii

is this just too much for 1 day to then have to take the ferry to Capri?

Probably.

It will be a rushed 'run through the ruins', or so short as to leave you wishing you had spent more time.

Set aside a day to do Pompeii and anything else you do that day is a bonus.

My $0.02

I am doing a similar trip from Rome to Amalfi Coast. I had your same question about timing and luggage. After doing some research we decided to do a private car from Naples Train station to Amalfi. Check out daytrip.com private car service and you can purchase a stop in Pompeii and leave your luggage in the car.
For me, we are not willing to dedicate a whole day to Pompeii, so it was a couple hours or nothing so we chose a couple hours.

I haven't used it myself but dozens of posters here have used the staffed bag check at the Pompeii Scavi entrance with zero complaints. They say there is also checking at the Circumvesuviana station, but I recommend the Pompeii ticket office area, only 250 feet farther.

You need to read up on massive crowding on this no-reservation commuter train. It can be hard to enter the cars even without luggage.

I agree that 2 hours is rushed for such a big site. I trust you know you can ferry to Capri in either Naples or Sorrento. Neither marina is right by any train station. Sorrento is a shorter ferry ride, and has a public bus to the marina. Fixed rate taxi in Naples could be convenient.

As Tim said. We did Pompeii in a couple of hours but again, it was rainy but enough for us. Leave luggage at the Pompeii entrance.
Ferry to Capri from there.

Way too much. With three kids in tow and luggage, go directly to Capri. Too many transfers to enjoy the trip. With five nights in Capri you have lots of time to schedule a day trip to Pompeii. Good luck ------

I've never been to Capri, much less been there on the same day as Pompeii, but I'm not sure this is such a clear-cut situation. As noted, there's luggage storage right at Pompeii, so the luggage will probably only be an annoyance on the Circumvesuviana--unless the timing works well for the Campania Express or the family decides on some sort of private transport between Naples and Pompeii--which seems a good idea for a group of 5 people. The ages of the children would be a factor here. I wouldn't want to tackle the Circumvesuviana with 2 or 3 kids too young to manage their own luggage, that's for sure.

On the other hand, if Pompeii is treated as a day trip from Capri, that's an extra round-trip ferry ticket for each person, and there'd be the business of getting from the lodging place down to the ferry port, and back up at the end of the day. I don't know whether the kids will qualify for lower fares on the ferry, but it appears the current standard one-way fare is from 16.50 euros (85-min. trip) to 25 euros (50-min. trip). I have no idea whether fares will be higher in late May, but it wouldn't surprise me.

There's also the matter of paying the very high lodging rates on Capri for a day when you're mostly not there. That's something I don't like to do.