You can certainly get more autonomy for the EIC as the result of civil war in the UK, though it's much harder (if not impossible) to have full independence. As it may be pertinent to the thread, there's an explanation of the government of India as established in 1833 (and due for renewal in 1853) located
here.
I think the best way to go about it would be to have a relatively long military campaign in the UK itself between Chartist and Royalist forces. Just to nitpick, though, I think it more likely that you'd see a broader opposition with working class support (like the 1832 reform movement) rather than straight Chartism. The opposition would probably aim to replace Ernst with the Duke of Cambridge, who has a good claim to the throne and was by all accounts a rather affable chap. It's not impossible that there could be a separate working class rebellion against the opposition movement, as tends to happen in civil wars (e.g. the Levellers, the Paris Commune, the Bolsheviks).
In the end, Ernst August loses and heads to the Continent. While he decamps to Hanover with the remnants of his supporters, he sends a new Governor-General out to India. I suspect that if he has at least some MPs and peers with him, or members of the board of Control, this would be viewed as the legal succession. It doesn't seem likely that the original Governor-General would be prepared to hand over to a Radical Republican London appointee, particularly if the original is an Ernst August favourite.
The new governor-general would have instructions to do whatever he needs to in India, as long as he can send back regiments and money to support the Royalist cause. This includes raising additional EIC European regiments to free up British infantry, to be achieved by widening the recruitment base beyond British nationals. This would create de facto foreign legions, as was done historically in the Napoleonic Wars and Crimea. I'm not sure whether there would be Chartist agitators among the British infantry overseas, but I suspect it wouldn't be a major issue. If there was a mutiny, as long as it's minor, a tactful governor-general might just grant clemency provided those who aren't ringleaders transfer to the Company's army and serve the rest of their careers in India.
If I recall correctly, the 1833 charter lifted a previously-imposed colour bar, so it would have been possible to employ talented Eurasians in the lower ranks of the civil service. The issue of whether Eurasians can be promoted to higher ranks would only emerge so much later, as officials reach the end of their careers, that the EIC would either have found an alternative source of recruits, or would have reconciled itself to mixed-race individuals being employed.
(I'm aware that a certain amount of the above is unrealistic, but then so is the idea of anybody liking Ernst August enough to die for him in battle.)