If it were mine I would first ascertain the worth of the piece in both dollars and sentimental worth to you. There are many companies that will sell you restoration curved glass panels and many more that will also do the work.
Depending on the value it may well be worth doing it correctly and making the piece worth more than the cost to do it. If it is deemed not very valuable but something you enjoy and you are ok with a poly replacement then go ahead with the DIY approach. The glass was held in with thin wooden strips that need to be removed and re-nailed after the plastic is bent into place.
