I was in the market for a new bow this year, and between the two, I'd favor the Invasion.
I was lucky enough to have friends/family with several different bow models, as well as the shop test drives. I also used to have a z7 (standard, not "Extreme"). I test drove over a dozen different bow models
Frankly, comparing a single cam with a binary isn't really apples to apples anyway, but that's a totally different issue. In my eyes, the level nock travel and self tuning component of a binary cam is a superior design, so even if everything else were equal (which the z7 is NOT equal to the Invasion), I'd prefer the binary Invasion.
Personally, I bought the Destroyer 350. Money wasn't an issue, and the Destroyer was only $100 cheaper than the Invasion anyway, but honestly, I felt like it was a smoother draw than the invasion, and a better balanced bow. The Destroyer and Invasion both have the Bowtech "Overdrive Binary Cam" system, but for some reason, the Destroyer feels harder, but smoother, than the Invasion. It's also a faster bow. I'm using blaze orange and black tiger bohning arrow wraps and blaze orange and black tiger blazers, and under 30yrds, I can't see my arrow in flight. I have a Trophy Ridge vertical 5pin set for 30, 45, 55, 65, and 75yrds... This thing is FAST (28" draw is giving 338fps average with 384grn Gold Tip Pro Hunter 7595's, 100grn Rage 2 blade practice tips).
I also love a hardcam bow...
In my hand, the z7 feels heavier (even though it's only actually 0.1lbs heavier), and it doesn't balance forward for me at all, actually leaned back slightly. I use a fairly low "target" style grip position, and I have always needed a LOT of stabilizer weight on my z7.
The Destroyer is a very smooth draw, the only down side is that it's a hard cam that allows ZERO creep, and if you're not looking for that, then it's not the bow for you (nor is the Invasion). Obviously, the z7's claim to fame is it's smooth draw, but honestly, I don't think the z7 Extreme is even as smooth as some of the other single cams out there (Bear Strike, Diamond Fugitive). The Destroyer and Invasion are plenty smooth, with the Destroyer the smoother of the two, but the Solocam z7 (A MUCH SOFTER CAM) is admittedly smoother.
The Bowtechs are more efficient bows, and my arrow flies like a laser from string to target.