It would depend on who their local confederates are-- Brahmin advisors and Hindu generals (essentially just have the people who defected to the Delhi Sultanate OTL... not do that) pushes them one way, and qadis and Afghan tribesmen push them another. Who do they have regular contact with, and whose friendship to they consider essential to the maintenance of control?
If this successor khanate has no interests outside India, I think it will adopt some manner of Indian religion (Hinduism, Buddhism, whatever) in the manner of so many other Central Asian invaders of India.
- Its neighbors and trade partners will be Hindu. The Khilji conquest up to Madurai should have already occurred by this point, but the more enduring Tughlaq conquest that laid the foundations for Deccan Islam hasn't. The brief Hindu coup in the Bengal sultanate may also be more enduring.
- Most of this Indian khanate's population will also be Hindu, at this point the only solidly Muslim parts of India should be Sindh and bits of Punjab.
- The Muslims through the Gangetic plain can still reasonably expect to keep their customs under a Hindu administration, exiles from the west ranging from the Parsis to the Ismailis found accommodation in Gujarat. It's not a wholly desirable state of affairs but in the 1200s the population of believers will be too small and scattered to meaningfully contest it.
- While this Mongol administration will probably be faced with a powerful Hindu antagonist to its south like OTL Vijayanagar, it need not fear betrayal from within if it pays attention to Hindu interests.
If it does have interests outside India, like if it's also ruling east Iran or if it heavily prizes its links with the other khanates, it becomes more of a toss-up.