It will not be affordable to pick up the car in one country and drop it off in another. I have no experience renting cars in Europe and don't know how the economics will compare to train/bus transportation for 2 adults and 2 children.
I'll take a shot at what might be reasonable bare-minimum nights to spend in the places you've mentioned. For myself, I would want at least one more night in many of these places, and I suspect moving a family around at this pace will not go well. I do not have children, and this is all just my opinion, to give you (and other responders) a starting point.
I am attempting to allow for travel time between overnight stops in the figures below, but I have not taken the time to check everything on ViaMichelin.com . I am assuming you'll have a car in Slovenia and Croatia.
Venice: 2 (1 makes no sense; the arrival day is useless and Venice is expensive)
Dolomites: 3 (4 would be a lot better). No time for lakes.
Trieste: 2 (long slog from the Dolomites; 1 night would mean no sightseeing)
Julian Alps: 2 (3 to allow for some activities)
Bled Lake: could combine with the Julian Alps loop or visit from Ljubljana
Ljubljana: 3 (allowing for Bled; Ljubljana is a lively, family-friendly city with a castle)
Istria: 4
Plitvice: 1
Zadar: 1 (if you get not too late a start on the drive from Plitvice)
Split: 2 (no islands)
Dubrovnik: 3 (more for side trips to Mostar, Kotor, etc.)
If you can fit in Zagreb (bus/rail link to Ljubljana is handy if you're using two cars): 2
Since you aren't very focused on including Venice in this trip, you might look into flying into Milan instead of Venice. From some origins it is cheaper and/or easier.
Again: That's still somewhat rushed, especially for traveling with a family, and the trip is long enough that travel fatigue will probably be an issue, which calls for slowing down a bit. What I've laid out adds up to 23 days without Zagreb. You'll need the 28 days to include Zagreb and allow the extra days I've suggested in the Dolomites and the Julian Alps. You'll still have the issue of what to do about seeing Croatian islands, Mostar, and Kotor.
I don't think you have time for the Italian lakes unless you drop the Dolomites, and I suspect the Dolomite may be more appealing to the children.