The issue isn't so much the sensitivity (although that is a significant issue), it's all the other crap going on. You'll probably be able to filter out the Mains Hum, but every anything moving in the same axis as the test will confount the data.

I'm thinking we might set up the instuments near counterweight energy storage or pumped hydro, and some on the exact opposite side of the planet, and try to measure the movement of the earth that way.

We can already see a change in the length of a day after big earthquakes and dam construction/destruction, but I don't think the acceleration has ever been measured directly.