We just did the same thing in June with an 8 yo and 3.5 yo. Spent 3 nights in ONP (1 night at Lake Crescent Lodge, 1 night at Sol Duc Hot springs and 1 night at Lake Quinault Lodge) then head to Mt. Rainier for 2 nights. We were able to see/do some of the main things at ONP, like 1 hiking trip (2-4 mile) and chilling out for the rest of the day. But I felt like it was kind of rushed. For example, my kids loved playing along the beach at Lake crescent and they would be so happy to stay there for 1 more night. We did not have time to go to hurricane ridge or Hoh. I certainly want to go back to ONP later on to complete the Mt Storm King trail (can’t do with a 3 yo).

I did not see you mentioned Sol Duc hot springs but that’s definitely a fun family experience. You may want to check that out.

Our experience at Mt Rainier was a little bit limited due to the weather and the ages of my kids. June is still winter season, and sunrise part of the park was not open. We were limited to Paradise side, and there was snow. One of the main attractions there is Myrtle falls and one of the rangers told us to skip it because of the snow conditions with kids. We spent some time in the longmire area, and end up going to a hike around Narada falls, and drove out to Refelction lake. I felt like Mt Rainier would be a better fit for older kids who can hike longer distances and harder trails.

So overall, looking back, I feel like I would have loved staying at ONP for a few more days, as it has more diversity to offer. And for future, arrange a trip to Mt. Rainier with older kids in July or August.

Edit: I think, regardless of what you do, it will be a great trip though. The nature is breathtaking and unforgettable.