If I may make a suggestion, put the claims of the respective sects aside for a moment.
Like many entities in a modern society, religious organizations seek and obtain state recognition. The purposes are varied, but in my, perhaps cynical view, it comes down to tax and limited liability benefits. Nichiren Shoshu is a sect of Nichiren Buddhism espousing certain views, but it is also a corporate entity recognized by the Japanese government enjoying tax benefits and limited liability protections. Nichiren Shu is the same. Both of these entities operate as corporations do to further their purpose - their purpose happens to be the promotion of certain religious functions. They own and operate temples, seminaries, universities, missionary organizations, etc. etc. They earn money through donations as well as the management of profitable assets like real estate.
I hope I'm not destroying some image you have of these organizations in pointing this out to you. Religion is a business, just like any other. It might be the second oldest business after prostitution. By no means are Nichiren organizations unique in this respect - Buddhism in Japan is BIG business. To illustrate, the oldest continuously operating for-profit corporation in the world (with a government charter) is a Japanese construction company closely tied to the Buddhist establishment that specializes in building temples. Over 800 years old.
Now, the specific claims. There were no sects when Nichiren died. He had six senior disciples and almost all existing Nichiren lineages can be traced to these six people. I say lineages, because lineage is not indicative of present sectarian associations. Moreover, these lineages crossed lines at numerous points through the last 700 years.
When Nichiren died, his six disciples all returned to their home regions to continue their shakubuku activities. These home regions are all around the Kanto region of Japan - basically the area around modern day Tokyo and the adjacent provinces. Even though you can cross the entire region by train in a few hours today, in the old days, it was several day's journey, and particularly after the fall of the Kamakura government when Japan descended into the warring states period, travel was not that easy. The result is that you end up having a lot of variation as temples developed autonomously in relative isolation taking on idiosyncratic traditions tied to the local communities. Similarly, you had a lot of variation as Nichiren's teachings themselves were interpreted through the lenses of particular individuals and groups.
Also, shortly after Nichiren passed away, missionaries went to Kyoto and within a few hundred years a large Nichiren community had developed made up of primarily merchants and other townsfolk. This actually became the center of the Nichiren community for many decades and the Nichiren community in Kanto in effect became satellites of the Kyoto community, dependent on them for financial support. With the money also came the people who were appointed to the leadership positions in Kanto.
The warring states period ended in a total war that, without going into detail, resulted in all of Japan reunited under the rule of the Tokugawa Shogunate with its capitol in Edo, now called Tokyo. In the process, many of the main Buddhist institutions were completely destroyed. The Tendai center of Hiezan was burned to the ground - they say that as many as 50,000 monks were killed - and Tendai has never recovered. Nichiren Buddhists in Kyoto were genocided. Very few institutions escaped unharmed. To compound the problems for religious institutions, they came under tight government control, such that competition for converts was basically outlawed. Whatever temple a family was associated with at the opening of the Edo period was the temple their descendants would be tied to for the next 350 years. The Tokugawa Shogunate rebuilt Buddhist institutions, but always on their terms - they controlled the money for construction of temples, for supporting monks, etc. The vitality was smothered out of Buddhism in Japan, except in one respect - within the restraints of government regulation, there was one way to make money...
By attracting pilgrims. One advantage that the Tokugawa unification had was that travel became safe again. People started traveling and much of the travel was done within the paradigm of religious pilgrimage which has a long history in Japan. It makes sense. Many of the religious sites are awe-inspiringly beautiful. Resort towns sprang up all over. Money flowed if you could attract pilgrims. To attract pilgrims, you had to have something special and worthwhile to get them to come out of their way for you. Unfortunately, this pressure, I think, was one of the main driving forces behind the evolution of Buddhism in Edo Japan - not actual religious teachings. Oh, also funeral rites which to this day are extremely lucrative. Many people joke that Buddhism is nothing but a funerary business anymore.
With the fall of the Tokugawa Shogunate in 1867, Japan went through another revolution, the Meiji Restoration. With the Meiji Restoration, there was also another backlash against Buddhism. The reasons are complex, some related to the state sponsorship of Japan's indigenous kami worship, some related to the rise of neo-confucianism, some related to a modernization impulse, some related to the oppressive practices of temples in the late Edo period under the Shogunate's sanction. A series of laws were enacted to reorganize and control Buddhist institutions, and this is when you see sects like Nichiren Shoshu and Nichiren Shu established under religious corporation laws.
Some of the temples in Nichiren Shoshu and Nichiren Shu trace their lineage back to Nichiren's time and immediately after. The fact is, many temples, and therefore the sects to which they belong, have plausible claims to direct connection to Nichiren Shonin.
However, a lot of stuff happened between the death of Nichiren and the establishment of these sects. I don't know if it is possible to look back through that fog and judge that one sect that has a relatively modern beginning is more legitimate than another solely on the continuity of a particular temple or lineage.
For me personally, the only thing possible is to look at the legacy Nichiren left behind in the form of his writings and find lineage in that. To the extent that the people who make up the clergy and leadership in these various institutions are sensitive and thinking human beings and can offer beneficial guidance in one's own practice, I think an argument could be made that they are upholding the lineage.
If you want a single, definitive answer that this temple, or this sect, or this teacher infallibly has the most legitimate claim to Nichiren's heritage, I can predict with absolute certainty, you will, sooner or later, be frustrated or let down. Your enlightenment is yours to work out.
BTW, to correct a minor matter - Nichiren Shoshu was not founded from within Nichiren Shoshu. This was a lay movement that was closely tied with Taisekiji for a while, but has its origin outside of the temple institution.