I'd go like this:
Inquisitor: Can only be used on one of your own cities. Kills half of the population that doesn't follow your State Religion, and converts the rest. Round all fractions up. For each population killed, the city gains 2 unhappiness for 10 turns.
Let's say a size 9 city starts at (60% state religion, 40% other). 20% are killed (1.8, rounded up to 2 people), so it's now a size 7 city that's 100% religion A and has +4 unhappiness for a while.
(The "offensive" use for it is this: if you've just taken over a new city that's full of heathens, you can send in an inquisitor. He'll kill half the city, but now it'll be full of loyal people who follow your religion. No enemy will be getting shrine income from the old religion, and it should lower the chance of culture flips.)
The rest I think you could handle through new promotions. For instance:
Crusader: +20% strength versus units of a civ whose state religion differs from yours
Possibly add a series of these, like one that gives a bonus when attacking cities whose religion is different.