If zats were so great, did they ever explain why everyone didn't just use those? If they DID explain that (either in the show, or maybe in the AEG rpg), then you could build off those limitations. All I can think of is that they can't make more, so they're not going to replace their primary weapons with zats. But, I'm pretty sure everyone or at least more people than just Teal'c carry them as sidearms later on. If that's the case, then we can work with that.
1. Zats should be better than pistols.
2. Zats should have drawbacks to where they are not an obvious optimum choice over an AR or SMG with armor-piercing rounds (which is essentially how they treat the P90 and it's 5.7mm ammo).
I'd strike a balance as best I could between what's shown on the show and rpg playability. I think AstronautAndy's suggestion of treating them like juiced-up tasers works. Have them deal say, maybe Weapon:2 physical damage and apply "Stunned" to their target. With any decent enough attack roll, you should be able to deal enough stress to Take Out a mook or minion, someone who's not going to take Consequences. Then, you can use the free tag on "Stunned" to initiate a Compel for unconsciousness. If the target doesn't have any FP, they can't refuse the Compel and they go down. Now, when it comes to killing on the second shot? Honestly, that's where you have to leave off the simulation. If you Take someone Out after they've taken a hit from the zat, then they're dead, but there should be NO shortcuts to character death like "Second shot kills".
Limits? Make the zat short-ranged, maybe 1 zone at most. It depends on how wide your own concept of zones are and how far you let other ranged weapons fire. Make zats' range shorter than that. Make zats run out of juice. Make the DC to acquire them high, if that is even an issue in your campaign. But those are just delaying tactics - if the zat is in your game, eventually the PCs will encounter them and then they will get their hands on them.