score:7
Even if they increasingly tend to emphasize the cruise side of their business on their website (to the point that I cannot manage to find the relevant page on the global Hurtigruten website), you can book port-to-port tickets with both Hurtigruten and Havila (a new operator following a tender from the Norwegian government to renew Hurtigruten's contract to serve this line). If the crossing is long or overnight, I think they may still require you to book a cabin as well but you can certainly take only part of the journey or stop along the way before hopping onto the next ship.
Those are the only two regular services that go along the whole coast all the way to Kirkenes. Hurtigruten's sailing plan is here but Havila's ships will follow the exact same route on days not served by Hurtigruten (anymore). The only alternative (depending on the part of the route and season) is to piece together local ferries and busses (trains do not go even as far as TromsΓΈ, let alone Kirkenes). Visitnorway.com has a helpful page with some ideas to travel around Northern Norway.
Note that Norway is a relatively expensive country, it is going to cost a lot for a family, no matter how you slice it. From that perspective, a cruise is not necessarily a bad deal as cruise tickets are full board, you do get to see a lot and have enough time for a quick visit in key places. Port-to-port tickets or bus travel will give you some flexibility and the pleasure of planning your own trip but I would not always expect massive savings. As a new operator, Havila does seem to offer great bargains, though, so I would look into that in any case.