I'd also recommend Paulicians, one of the games I had most fun with was a Paulician Irish Merchant Republic. I started as an Irish tribal and gave my son to the Paulician Patriarch to convert him, once the son inherited I could declare holy war on my fellow Christians, but here comes the kicker, as you are also christian they can't declare HW on you, also raiding weakens your targets before getting HW'd so even if other christians join defensively, the wars are usually quick (you can easily siege down their depleted (from previous raiding) counties).
After that I formed a MR. As you are given the option to vassalize your religious head, I did so. The religious head is a secular feudal ruler and has primogeniture, if he dies without descendants, guess who inherits the Paulician Head title? YOU.
In the end I was a MR able to raid anyone, able to HW and GHW anyone, raking cash and destroying rival trade posts thru raiding. Rome, Constantinople and wherever I wanted was raided (it also allowed to take titles very easily through war), aside from that I had enough income to sustain a permanent mercenary army, free upkeep vassalized raiding holy orders (after overtaking apostolicism), staking traits (exalted, paragon, raider, icon-breaker, crusader, etc), even I took prisoner the Pope (and other high level people) several times...
All in all is like being a reformed pagan that can raid (CK2+ doesn't let reformed pagans raid) and can declare HW on his religious group but not the other way around.