In fact, many aspects of the first seder are also Rabbinical. However, there is no difference: everything done at the second seder is identical to the first seder (Shulchan Aruch Orach Chaim 496:2). Kiddush - the first stage of the seder - must be after nightfall (O.C. 472:1 - look up halachik nightfall for your location at myzmanim.com).
That said, the Maggid (story) section of the seder is not actually that long. Even with everyone taking turns reading it around the table, it doesn't take more than about 45 minutes. Telling your children about the slavery and leaving Egypt is the main mitzva, and you should do that in your own words if they are too young to appreciate the words of the Hagada. You could then quickly read through Maggid in the Hagada to yourself, which wouldn't take more than a few minutes.
When my kids were little, we made a big deal of the end of Maggid with the list of the 10 plagues (acting out the plagues, plastic frogs/farm animals/wild animals, hole-punch circles for dust/lice, etc). Then we got to eating matza and marror, fed them a little, and put them to bed.
The four cups must be drunk in order after saying their accompanying parts of the seder (i.e. the first cup is kiddush, the second cup after maggid, the third cup after birkat hamazon, and the fourth cup after hallel - O.C. 472:8). Potentially some sections of Maggid could be skipped (heard from Rav Yitzchok Berkovits) but it's not going to make a difference in attention span for little kids - if the question was teenagers it would be worth consideration.
If they are too young for to stay up (give them a nap in the afternoon!) then the suggestion of doing a model seder earlier is excellent.