1200W inverter in 2007 Tahoe

I have a 2007 Tahoe Z71.

Am I going to damage the alternator by running a 1200w DC inverter over a 5-7 hour drive (I wont be pulling the whole 1200W but I wanted to get one that wouldn't over heat.) My usual draw will probably be about 400-600 watts.

Thanks!

Oh! also I wired the Black Cable (4 gage) to the frame of the tahoe instead of the negative on the battery, can anyone see a problem with that?

12 watts at 12 volts = 1 amp. So lets say a maximum of 600 watts would be 50amp draw. These rigs come equipped with a 130 amp alternator if my memory serves me best. I dont see any problems with that. Just remember that every piece of electrical equipment such as inverters are only 95% efficient. So a 1200 watt inverter is going to constantly pull 5% of 1200 watts. .05x1200= 60 watts even when not in use. All in all i dont see 50 amps being a big problem at typical 60 mph, rev speed which is typically 1700rpm. You should be putting out 100 amps. Just keep an eye on the volt gauge. If it has a problem holding 14 or more volts, then it might be a bit much. I however havent had in problems with a constant 50 amp draw.