Exiled Sikh State

A rather random, whimsical question, but is there anyway to get a Sikh state "outside" of India region ? Say, in reversal to how Mughals came to India by Timurids being driven out of their previous homeland, maybe the Sikhs or at least one of the Sikh misls at least, could've been driven out of India and end up setting themselves up in, say, Central Asia ? Or perhaps it's easier to instead make the Mughals a colonial empire in which one of their colonies being a Sikh settlement and at one point break off from the home country or becoming something of a dominion ?

Since I just can't choose between the Sikhs and the Mughals

There's a large Sikh diaspora in western nations. While it's unlikely they'll try to carve land out of these nations, they may form a foreign movement for independence of Sikhs in India. Their best chances out of India would be to try to repopulate the Punjab region in Pakistan with Sikhs, as there his historical basis there. Problem is not a lot of Sikhs there. Another way is if Isreal repeated itself but with Sikhs, for instance, if there was a Sikh shrine found in let's say, Central Asia, people for what ever reason may declare it holy and start moving there. For a Sikh nation, google 'Khalistan'.To your Mughul question, yes I suppose it's possible. Yet typically they didn't force ethnicities out of their conquered lands. If they relocated them, or did something to make the Sikhs flee, then it is totally possible. Hope I helped!

Maybe if the British outsourced them to South Africa?

The easiest way for an'exile' Sikh state is probably to have secondary passports that establish something of a nation for the international Sikh diaspora; you could be Canadian and Sikh national (or international in this case) or an Indian and Sikh.

Another possibility is to have Akbar not be the Emperor thus butter flying Amritsar becoming a Sikh fief- if someone more orthodox comes along, it might even be persecuted earlier on, causing an exodus of sorts to another region.