if you set it at 700w
700w / 80% efficiency (varies depending on the amp, but that's about average) = 875 w draw
875w / 14.4v = 60.7 amp draw.
(1100w/ .80 = 1375 / 14.4 = 95 amp draw) That's when your amp is maxed out playing full volume. Not sure what vehicle you have, but a factory 150amp alt is on the higher side. My guess is that's a newer vehicle or maybe a work truck.
Your battery should be fine. It's a storage bank, and that should be within comfortable limits.
For an alt, when you are playing that at max volume (700w), you are drawing 60 amps. Between those notes, and songs its drawing much less so that it could recharge. Your vehicle is only drawing it's max, when you have your ac on, electric fans on, lights/brights on, charging ports/cig lighters/ dash lights, turn signals, brake lights. Other than that you aren't even close to using up that 150amps.
I don't think you will need to, you certainly don't need to upgrade either one before testing it out. If you play that on full volume for hours on end, and then notice your battery voltage start to drop, THEN your alternator can't keep up, and you need one that can charge more. If it was me, having a factory 150 amp alt, I wouldn't even bother with either until I KNOW there is a problem. I certainly wouldn't expect a problem with that.
(if you do plan on upgrading the alt, YOU NEED to think about the rest of the system you MIGHT want to upgrade in the future. Then would be the time to buy that 300 amp alt, and not after you buy a 200amp alt, then decide you want another amp. Cheap alts are $200, and good ones are $500+. Don't buy a $450 alt, then in 6 months need a $550 alt).