Unfortunately, the lockjaw only appears when you kill a certain hoodlum from the second wave. Here's what I have done so far (updated version

):
-Kill the first three hoodlums in combo with a yellow gem. (670 points)
-Take all the remaining gems, kill the caporal on the ground, get the lockjaw, head towards the flying hook, turn backwards and kill the second hoodstormer, jump so that the third hoodstormer appears and kill it, jump onto the balloons, kill the second caporal on the upper platform, get the gem on the last balloon and the four gems on the tower. (3500 points)
The main advantage is that I kill the third hoodstormer before jumping onto the balloons, which prevents it from shooting at them. However, it's quite tricky and requires that I kill the second hoodstormer as late as possible. It's really hard at first but now I do it almost all the time.
You may think it's possible to link both combos, and well, I think it actually is, if you're quick enough to kill the second hoodstormer in combo with the first wave, but without this hoodlum, it's almost impossible to get the third hoodstormer in combo as well (or maybe you could use the caporal on the upper platform instead?), which would result in a loss of points.
There's way for improvement though: it's possible to get 180 bonus points by getting the gem on the seventh balloon before killing the last caporal, but I can't be bothered (the whole combo is tricky enough), and we also have the possibility to include the gems in the water. For example, I could start the second combo with them, then kill the caporal, but if I collect the gems on the ground at this moment, I won't have the time to head towards the flying hook so that I can kill the third hoodstormer in combo with the second one. It's not so much of a problem anyway, because I can include these gems in the first combo.