Here is what The Man is Seat 61 says about traveling by train with kids:
https://www.seat61.com/european-train-travel-with-children.htm
I would definitely look at the age limits on the local transports in each city, as they may not charge at all for the kids or offer a family pass or whatever. I'm not an expert.
A great activity for kids in Copenhagen is to hunt for Thomas Dambo's trolls. thomasdambo.com There are loads of blogs out there talking about how to get to the more remote ones via public transport and bikes. We did it with a car and had a blast.
In Oslo, be sure to see Vigeland Park's amazing sculptures, the Viking Ship Museum (accessible by bus or ferry) and the modern art museum (beautiful building).
In Bergen, you shouldn't need a transit pass. Everything is in walking distance. I mean, there are lots of hills and steps, so the 3yoa kid won't be able to walk far, but I'm not sure the bus helps. I guess it depends on where you're staying. The Floyen is amazing. I would allow a whole day up there. There's a massive play area for kids with ropes and a zipline. There are restaurants up there, but you could also pack a lunch. Incredible views for your picnic.

Consider travelling Copenhagen-Oslo on the appreciated overnight ferry, much loved by North Americans.
Two countries, three counting the short Helsinki stop, is satisfactory with the amount of time available. For more amusement park fun try Bakken amusement park (free entry) at Klampenborg just outside Copenhagen, with a lovely family-friendly seaside beach. For a taste of sweden consider Malmö, our prominent third city, an easy commute from Copenhagen (37 min). Try the long family-friendly Ribersborgsstranden beach, or the charming Västra Hamnen bathing jetties (deep waters, ladders only), with views and gelato.

Well, a car round trip from / to Copenhagen would be easier in terms of rental car which would give you more freedom and the chance to overnight in family cottages or cabins on the countryside. Also some interesting destinations in the country would be nice for the kids. The loop would be as described by you and from Bergen back to Copenhagen either via ferry Oslo - Copenhagen or Kristiansand - Hirtshals (DK). 5 children seats would be a little cost issue.

There are some worth seeings that are much easier to reach by car:

• World Culture Heritage cave carvings in Tanumhede
• Moose farm on the way Bergen - Kristiansand (S)
• Walk-behind waterfall Steinsdalsfossen
• Viking village in Gudvangen

If you decide for public transport the itinerary looks OK because there are not so much options. Just to mention one. If you stay 3 of 5 nights in Voss and not in Bergen you could explore Hardangerfjord area by rental car from / to Voss.

Good news is that all Scandinavian countries are very kids friendly. You can also stop somewhere at a tourist info, a gas station or a supermarket to ask for using the toilet. You will find most of them clean.

Restaurants are often more expensive, esp. at the tourist spots. Supermarkets are cheaper at fjords, large cities or near large streets starting with "E" (Europe streets).

I personally feel like 5 days is too much in Bergen compared to the rest of your itinerary unless you are REALLY into hiking. Three would be good. I would give the two extra days to Copenhagen personally, or maybe distribute them between Copenhagen and Oslo. How long are you planning to stay in Oslo?

The Viking Ship Museum in Oslo is closed and not due to reopen until 2027, when it will be renamed the Museum of the Viking Age.

There is incredible fjord scenery accessible by modifying your plans for travel between Oslo and Bergen. Rather than taking the mainline train straight through, you could follow the Norway in a Nutshell route as follows:

• Mainline train Oslo to Myrdal (quite scenic)
• Flamsbana train Myrdal down to Flam (on fjord--very scenic rail line)
• Naeroyfjord ferry from Flam to Gudvangen (one of the country's most scenic fjord trips)
• Bus from Gudvangen up to Voss (scenic)
• Mainline train from Voss to Bergen

Connections are easy, but it makes for a long day. Unfortunately, lodging is much in demand in Flam (timewise, probably the logical breakpoint), and you'd have to be really, really lucky to find accommodations at this point that are accessible without a rental car. You might find something in Voss, which is way, way past the midpoint of the trip, but that would allow you to end your day somewhat earlier than if you traveled all the way to Bergen. Voss, while not located on a fjord, is a center for active sports, so there might be things you children would enjoy doing there.

NorwaysBest.com sells Norway in a Nutshell packages, but it's easy enough to buy the component tickets yourself, and the savings would be substantial for your large family (I'd guess around $100 per person). You'd buy Oslo-Flam (through ticket covering train and Flamsbana) from Vy.no, paying via PayPal. NorwaysBest sells tickets on one of the Naeroyfjord ferries; tickets for the second company's ships are sold on the company website, lustrabaatane.no. For the Gudvangen-Voss bus you'd pay onboard via credit card. The Voss-Bergen ticket you can purchase from Vy.no.

The Oslo-Myrdal rail fares vary, increasing as the travel date approaches, and those trains do sometimes sell out. Whether you opt for the NiN route I've outlined above or decide to go straight through to Bergen, you should consider buying those tickets early. (Check rules regarding changes/refunds.)

The Flamsbana can also sell out, I think, though I believe the fare is fixed. This is another reason to buy the Oslo-(Myrdal)-Flam ticket early if you opt for my scenic-route suggestion.

I don't know whether there's a risk of a sellout on the Naeroyfjord ferry. There are several ferries a day, but when you try to string together the components of the NiN trip, your options narrow a lot.

The bus (which is Skyss Bus 950) doesn't seem to be a problem, capacity-wise.

The onward train trip from Voss to Bergen runs frequently and I think has no reserved seats, so there's no rush on buying that ticket.

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I agree that five days in Bergen is a lot. You could definitely use more time in Copenhagen, and Gothenburg is also very nice. You don't specify how many nights you plan to be in Oslo.

Bergen is a beautiful, if hilly, town. I loved the art museums and just walking around, looking at the architecture. As previously mentioned, there's good hiking at the top of the Floibanen funicular. I skipped most of the historical museums because they just didn't sound very interesting to me. Even Rick says something to the effect that they're mainly interesting if you take the included tours. I wonder whether your younger children would be bored, given that I thought I probably would be. So I don't know that a sightseeing package in Bergen will prove to be a good use of your money. Everything in Norway is very expensive; you might prefer to spend your money on scenic transportation (like the lift in Bergen and the Naeroyfjord ferry) rather than what I perceive to be marginal indoor sightseeing.