I highly doubt the Mongols would let a family who claim to be the equal of the emperor of China live.
The Mongols weren't going to let the Japanese tenno claim to be an equal of the Mongol Khan, but as long as he admitted vassal status to the Mongol Empire I think the Mongols would probably leave him alone. The most important thing as far as the Yuan Dynasty was concerned was for Japan (or Korea, or Annam etc.) to follow the basic protocols of Mongolian vassalage: tribute, hostages, and supplying men/materiel for war. Anything else was pretty negotiable. (Of course, whether the tenno would accept such demotion is another question entirely...)
Kublai Khan, in the beginning, was quite willing to treat the surrendered Song Emperors kindly (to the extent that Song forces under Li Tingzhi were unable to convince Empress Dowager Xie to flee to the fugitive Song court in Fujian), but that changed once the two youngest Song members were spirited out of Hangzhou and became rallying points for Song resistance, which continued all throughout the Yuan Dynasty. In such circumstances the continued existence of the House of Zhao was going to be problematic.