This is a very hard scenario to go through with and to achieve it you may have to change the very meaning of Mughal. What I mean by that is it is almost impossible to further the amount of localisation the Mughals underwent without not making them the Mughals but just some other Islamic dynasty ruling Hindustan.
Without going into too much detail, by the time the Mughals were entrenched in India so was Islam, nearly all of Sindh Kashmir and at least half of Bengal being Muslim with large pockets and enclaves throughout the north. Coupled with the fact that until Akbar's treaty with the Rajputs there was NO place for Hindus (hell, even Indian Muslims!) in the Mughal feudatory system I doubt this can be achieved. Since the only people associating with the Emperor would be his Turkmen kinsmen and loyalists, Afghan officers, Persian artists and courtiers and Arabian traders many Mughals (excluding Akbar) would not even talk to Hindu sages, much less get them to convert!
Next one must view the mindsets of the House of Gurkani itself. The thing about the Mughals was that they considered themselves above all other families and Lordlings in the entire sub-continent and their domain. Why? Because they believed they were born from different stock, Chengeizī Khūn, as one would put it in Urdu. In English that means 'Genghis' Blood'. That also inadvertently made them paternal descendants of Timur, a man who had left so deep a psychological impact upon north India and the Middle-East up till the Levant that simply uttering his name was said to silence the entire court according to the Baburnama. In their minds they were belonging to a 'race' unlike any other and given that Mughal is a bastardisation of the word Mongol, the few thousand Chagatai Mongols that came with Babur to India found themselves more akin to the eyes of commoners than simple rulers. They were expected to be better at everything than the masses of their Indian subjects, Afghan soldiers, Persian tutors and etc. The only apt comparison I can find is the Targaryens from ASOIAF (if one can replace Dragonlords with Horselords).
The reason something like the Din-I Ilahi can even exist is because the Mughals considered themselves far above all they ruled. Even the Safavids in Iran, also of Turcoman descent, tried to bring some sort of religious syncreticism between Sunni Islam, Shia Islam and Zoroastrianism under Mohammed Khobanda but it fell apart within a few years. The only reason Akbar could suggest something so audacious to the Ulema was because of the immense amounts of power the House of Gurkani held even in their early days. And it was also pretentious of Akbar to do so, after all he was supposed to be the defender of the Islamic faith and been Ghazi since the age of 12 when he had to knife a dying and twitching Hemu. However many Mughals considered themselves above such things and did as they pleased.
And there have been Mughal princes in the harem who have had both a Muslim and a Hindu naming. Jahangir was known as to his Rajput grandfather and mother Prithopal, a Rajastani version of his Persian name. Jahangir was quite open about his Indian heritage and merged it with his Mughal one, furthering them in the eyes of the nobility and peasantry.
Some (probably most) of is may seem disjointed since it is quite late here and I am writing this a bit groggy but I hope my argument at least ,Ames a little bit of sense.