Couple of things you can do:
Take one of the pieces and let it float at the top of your tank near the light. It will grow and form a bigger piece. Or just float all of it and let it grow.
Take a piece and put it in a bowl with water. Cover with another bowl and put it in indirect sun/light. It will grow and make a bigger piece.
IMO I'd try one of those methods to grow some before I'd tie to to driftwood. I've had it totally die on me after doing that, and I've had only a small section die and regrow.
You can try using a hairnet to anchor it to driftwood, but IME hairnets are dangerous in a fish tank. Your fish can get caught in it. You could try this in a tank that has no fish in it.
I would not try to use plastic mesh at all. The leaves are too big.
And yes, I do have some pellia that "sunk". It's all under the vallisneria and I forgot about it when I originally posted. I also have some that has grown almost all over a piece of large driftwood in my tank. IF you can get it going, it's great stuff. Just very delicate.
My 2 cents
Roan