There is a machloket among Tanna’im whether the cutoff between the times of Mincha and Ma’ariv is nightfall or plag haMincha (a proportional hour and a quarter before night).[1] There is also a post-Talmudic machloket whether night begins basically at sunset (Gra) with bein hash’mashot (halachic twilight) or around an hour later (Rabbeinu Tam). Our communities primarily follow the Gra that sunset is the main cutoff time. Thus, we regularly daven Mincha until sunset and Ma’ariv somewhat afterward.
The gemara[2] is flexible regarding the earliest time for Ma’ariv, saying that one can follow either opinion in the aforementioned machloket Tanna’im. The Shulchan Aruch[3] says that one should be consistent about which opinion he follows. Those in places that extend Mincha until night (which begins with sunset, according to the Gra) should not daven Ma’ariv during that time, barring a situation of specific need (she’at hadechak). Therefore, it seems problematic to daven both Mincha and Ma’ariv during the hour before sunset. However, there is more to discuss regarding your question.
The Shulchan Aruch[4] says that at the beginning of Shabbat we daven Ma’ariv earlier than during the week. The Mishna Berura[5] explains that we rely on the lenient opinion regarding the earliest time for Ma’ariv in order to gain the halachic benefits of accepting Shabbat early. The Magen Avraham[6] suggests that because of the changes on Shabbat in the service in the Beit HaMikdash, which Ma’ariv mirrors, there is an inherent reason to daven Ma’ariv earlier. If we accept the concept that Ma’ariv can be done before halachic night, we should permit it after plag haMincha. The Mishna Berura[7] cautions that on a day that we daven Ma’ariv after plag haMincha, we should be careful to daven Mincha before plag haMincha, to avoid a contradictory situation (tartei d’satrei). Even though he raises the possibility of allowing tartei d’satrei in certain extenuating circumstances, especially on Shabbat, he objects to it being as early as we are discussing.[8] (Some communities admirably arrange their early Shabbat schedules to avoid these problems.)
Let us move on to practical halacha. We believe that if starting one’s Shabbat early during the summer brings significant enhancement to the day, then this is sufficient grounds to do so. In addition, if this is the only practical way to meaningfully include one’s children in the celebration of Shabbat evening, this is reason to prefer, if not require, doing so. The parents should decide what works best for the family. Furthermore, realize that there is an old minhag to make Shabbat Ma’ariv relatively early and a widespread minhag in many communities to do so from plag haMincha. Just as we take a minhag of strictness seriously, so too we take one of (or containing elements of) leniency seriously.
Your idea of accepting Shabbat (after having davened Mincha), eating, and joining a late minyan for Ma’ariv will solve some time-related problems.[9] The Mishna Berura[10] says that one may do this if he starts eating earlier than a half-hour before night; otherwise, he should wait to daven Ma’ariv.[11] (Other ideas for leniency in this matter are beyond our scope). The Kaf HaChayim[12] cites the Arizal’s kabbalistically-based objection to switching the order of Ma’ariv and Kiddush. To us, however, a greater issue is that since our communities already have an established procedure for starting Shabbat early, altering it can potentially be divisive. If you want to use your system discreetly, you have the right to do so, but we do not want to encourage new minhagim.