In the later Kamakura period, there was a tendency for the shogun to be members of one of two lineages in the Imperial Court. Let us say that the alternating lines of succession, through death, agreements, etc. ends up with the Daikakuji line providing the shogun sometime around 1301 (perhaps Go-Fushimi of the rival Jimyou-in line is permitted to remain Emperor and a Daikakuji prince becomes shogun as compensation). If we look at the eligible candidates, among the most eminent is certainly Prince Takaharu (born 1288), who OTL would become the famed Emperor Go-Daigo of the Kenmu Restoration. He was the second son of Emperor Go-Uda from a mother who was not the most prestigious and was never expected to inherit the throne, thus making him the sort of man the Houjou regents might seek to install as shogun. Although he'd be only 13 years old at this point, it is very possible he developed many of his views because his immediate family and his future was so caught up in all manner of scheming and broken agreements. Let's assume he turns out more or less the same person as OTL--a scheming visionary.
One can easily compare the situation of the shogun in this time to the situation of the emperor--they were more or less powerless figureheads. However, Go-Daigo came to power OTL in part because there were positions in both the palace and Kyoto he as Emperor could influence. I'm assuming there was something similar for the shogun in Kamakura, and no doubt as shogun he would make endless visits to shrines and temples which is where he could gather allies from the clergy.
The opening Go-Daigo would have to regain the powers in the shogunate is the sheer amount of disorder by the early 14th century, given the Houjou regents themselves became increasingly divided and relied on their ministers to settle disputes (particularly after the abdicated shikken Sadatoki died in 1311). For instance, one member of the Houjou clan, Munekata, killed the very prominent Houjou Tokimura in 1305 and then himself was killed in retaliation. There were situations in the 1310s and 1320s when the shikken ruled for only a few weeks or even days because he was so disliked by segments of either the Houjou clan or prominent ministers like the Nagasaki clan. Likewise, he could still find allies in the Imperial Court--for what little that would be worth--given his family lineage. Provincial allies from OTL like the Ashikaga, Nitta, etc. still have much to gain from the weakening of the Houjou clan, even if they themselves will seek to install one of their own as Takaharu/Go-Daigo's successor.
Basically, how far could Go-Daigo get as shogun, and what sort of rule might he try with the Imperial court? He did consciously emulate Chinese history from his studies of Confucianism, Chinese history, and Chinese texts on governance (i.e. the era name "Kenmu" was chosen to copy the era name of Emperor Guangwu, restorer of the Han Dynasty), but I think becoming emperor would be a step too far. But certainly he'd try and centralise the state like OTL, and that would dealing with the Imperial Court.
One can easily compare the situation of the shogun in this time to the situation of the emperor--they were more or less powerless figureheads. However, Go-Daigo came to power OTL in part because there were positions in both the palace and Kyoto he as Emperor could influence. I'm assuming there was something similar for the shogun in Kamakura, and no doubt as shogun he would make endless visits to shrines and temples which is where he could gather allies from the clergy.
The opening Go-Daigo would have to regain the powers in the shogunate is the sheer amount of disorder by the early 14th century, given the Houjou regents themselves became increasingly divided and relied on their ministers to settle disputes (particularly after the abdicated shikken Sadatoki died in 1311). For instance, one member of the Houjou clan, Munekata, killed the very prominent Houjou Tokimura in 1305 and then himself was killed in retaliation. There were situations in the 1310s and 1320s when the shikken ruled for only a few weeks or even days because he was so disliked by segments of either the Houjou clan or prominent ministers like the Nagasaki clan. Likewise, he could still find allies in the Imperial Court--for what little that would be worth--given his family lineage. Provincial allies from OTL like the Ashikaga, Nitta, etc. still have much to gain from the weakening of the Houjou clan, even if they themselves will seek to install one of their own as Takaharu/Go-Daigo's successor.
Basically, how far could Go-Daigo get as shogun, and what sort of rule might he try with the Imperial court? He did consciously emulate Chinese history from his studies of Confucianism, Chinese history, and Chinese texts on governance (i.e. the era name "Kenmu" was chosen to copy the era name of Emperor Guangwu, restorer of the Han Dynasty), but I think becoming emperor would be a step too far. But certainly he'd try and centralise the state like OTL, and that would dealing with the Imperial Court.