You know, when I first read your post, I thought "I wonder if she is in the desert or something, because that's the only way I can see compost needing any attention." Well, then.
1) Add water to your compost. You could just water it like a plant (turn the hose on it every now and then), but even better, dump cooled cooking liquid onto it or something, there's no waste there. Like, if you make a pan of macaroni and cheese, save the water you drain, let it cool a bit (so you don't boil the microbes right out of the compost - though it doesn't have to be COLD, just cool enough to touch), and dump it.
2) Cover your compost (with a tarp or something.. heck, even newspapers?). This reduces the evaporation. If you think of it, uncover it before it rains then cover it back up after.
I think that's all you need to do. I would try this before trying the worm bin. I LOVE worms, but as a worm-killing veteran, I don't think they are as easy as a compost pile. You're going to have to do at least that much work for worms, so at least give the pile another shot.
All the work you described doing, with turning and adding more of this and less of that? Yeah, I don't bother. But I can get away with it because I'm in the moist Northeast. It just magically does its stuff. But for compost to be alive, it has to have water.
If your pile is reasonably moist, life will find it and flourish there.
ETA: Sorry, I remembered you have a tumbler. That should help reduce evaporation just fine, though it does mean you have to add water (since any rainfall is not going in there). How dry is your stuff? I assume it's pretty dry, or otherwise it would be teeming with compost life. The other possibility is that your tumbler has too small mass to really get going. Would you consider an open pile right on the ground? If not, that's cool, but just an idea.