Somewhere on Reddit,
@lambda x.x said that the best thing one can do in a one-culture run (and therefore, a one-faith) is to conquer fast, and I couldn't agree more. The faster you conquer the world, the more time you have left to convert it without stressing about modifiers like "religious zeal." With the addition of monuments, religious conversion has become rather trivial. So my advice is to pick a country which can conquer the world the fastest.
And I would advise against Sunni. I am currently in the middle of a Turkish one-culture attempt with the Ottomans/Caliphate, and Sunni feels underwhelming (and Sunni Ottomans were a mediocre pick for this tbh). I did a Hindu Mughals one-faith earlier, and it was much easier due to CCR stacking from Mughal traditions, Hindu monument, and the personal deity (75% CCR total). Hindus also get access to Buddhist monuments, which provide several missionaries and missionary strength bonuses — you can get something like 28% missionary strength (the trick to get Buddha is to not select a personal deity for a new ruler. If you don't select a personal deity, the game will eventually give you an event where you can pick Buddha). I also attempted a Byzantium one-faith Orthodox run and quit the game in late 1700's just because I realized I couldn't core/annex everything on time. The problem wasn't the conversions but the rate of conquest. As far as the achievement, I did it a couple years back as Austria. Again, the bottleneck of conquest was resolved with vassal feeding and one-click HRE integration.
If this Turkish one-culture doesn't pan out, I plan to do a run as Castile. You can kill all the colonizers early and reduce the potential heretic provinces in the colonial regions, and you can secure expansion routes with no-CB Byzantium and just 2 ideas from Exploration to land in Africa and SEA. Then you can form Sardinia-Piedmont and Italy for admin efficiency and CCR, which, again, speeds the rate of conquest. Any Christian denomination would work. It probably wouldn't be too hard to even become Hindu Italy, but I am not that great of a player and it's too much work for me.