Today I transferred the compost I've been making all summer from my compost bin into my holding bin. I found that a neighboring tree's roots had invaded the compost, creating a thick mat of tangled roots. Does this result in less nutrient-rich compost?

I placed some plastic trash bags on the soil surface inside my now-empty compost bin in the hope of preventing that from occurring again. But I realize the worms won't be able to lend a hand with the process. Does this sound like a viable solution?

Yes, the tree has been consuming the nutrients in the compost, so the compost is less nutrient rich.

Your compost bin won't work quite right with the plastic on the bottom. Mainly it will retain too much water and turn into a soggy mess. I think you need to either cut the roots back or move the bin.

I agree with @Henderman. I would probably move your compost pile.

Consider moving it into your garden. If you're composting vegetable scraps consider vermicomposting that. I don't have a "compost pile" any more, haven't for decades. Compost is only like 1% nitrogen because it bleeds off, and if your neighbor's tree roots aren't getting that, then your grass is. Weeds and veggie scraps go directly to the worms and woody stuff gets chopped up and used as mulch. A woodchipper is a wonderful thing - a thing I don't have one of anymore but will again post this next (and hopefully the last) move. I use lawn trimmings and other compost-ables, cardboard, and mulch in my garden beds, everything else is food for worms or the wood chipper.

BTW any good recommendations for a new wood chipper? LOL!

BTW BTW - I forgot. You can vermicompost right in your garden. Look up worm towers, I'm pretty sure I first read about them somewhere on these here forums years ago. I'm a potter and hope to start making my own post this next (and I can't say it often enough, hopefully LAST) move, but you can make them out of PVC. NO need to have messy worm bins or a compost pile. I don't think I'd put weeds down a worm tower but pretty much anything else I'd feed to worms would be ok.