When it comes to Economy of means achievement, I have found something.

I tried final level (chapter 5) just by itself, wanted to practice before the real run. Then I started real attempt. After finishing chapter 4, I got the achievement. I was quite surprised. My theory was later confirmed by user "Cireme". The game counted my first practice run on chapter 5 and then combined it with the rest (where I had like 85 grabs).

That means you can do the worst chapters first (4 and 5, since that crane is a bit anoying part) and then do the rest. Worst case scenario - you restart few levels instead of whole game.

Hope this tip helps you in getting the achievement.