Warning - contains spoilers.

After ep 11 and 12 I think I have finally more or less pieced together Li Mao and Li Lejun's relationship to the imperial family.

The big hint is in Li Mao's name itself. As I posted earlier on:
Lǐ Mào/李茂 was clearly named with a sense of humor as it is a near homophone of límāo/狸猫 or civet cat. Of course that immediately brings to mind the Chinese idiom límāo huàn tài / 狸猫换太 or the crown prince replaced with a civet cat. Or more broadly it refers to an imposter. For those who watched TROP, Ning Yi's nickname for Fang Zhiwei was little 小狸猫 or little civet cat/imposter. It originates from the story of how Emperor Renzong of Song was swapped out at birth by the empress.

In the context of KTTPH, Li Mao or Prince Qi is only a second or third rank prince bearing the imperial Li surname. The title Prince Qi has passed on to Li Mao, his fief is Qizhou which is not a rich fief, therefor he is from a relatively powerless branch of the imperial family. That is also why his sister Lejun is only a Xianzhu or a county/third rank princess. But according to Li Fuhua, he is the illegitimate son of the late emperor's Consort An and the Grand Tutor. The thing is, Consort An and the Grand Tutor did not dare to pass him off as a son of the late emperor. They instead swapped him out with the son of the late Prince Qi, Lejun's father. So Li Mao and Li Lejun were raised as true biological siblings and as far as they both know, that is what they are. It is only when their "parents" died that they were brought to the capital where they were more or less ignored by the current emperor and the ED. Li Mao is the civet cat that was swapped out with Prince Qi's son (maybe the baby was sickly and died) and now must earn his way up from the appointed position of Regent to Crown Prince.