We have a name for Shia people who live in the Caucasus. They are called Azeris. The Caucasus were the Safavids "home ground" in a sense. Due to how religious conversion and ethnic assimilation work together, groups converting to Shia Islam would be under a huge amount of pressure to become Azeris too. And if they resist assimilation, its up to their children will have to make the choice, and then their children and so on. That's the perspective an outsider would have. Being Shia and being Azeri would be seen as the same thing. It would be very difficult to untangle this type of thinking, cause it is after all partially true.
Modern day Dagestan, being one of the most ethnically diverse areas in the world for its size, IMO would be very vulnerable to this type of assimilation. TTL Dagestan would probably just end up part of an expanded Azerbaijan.
I don't know if this would work for the Chechens. It definitely won't work on the Circassians. I am not an expert on when or how Sunnism arrived to the North Caucasus. It seems to me though that the Sunni population would resist the idea becoming Shia as much as becoming Christian. You would need an much earlier POD than the Safavids.