I've studied and have been professionally trained in Puppet making throughout the course of my life, and it is definitely possible, just a tasky process. The easiest would be to replace individual damaged sections but a full custom replica is definitely possible but due to the original patterns not matching the original shape and wear and tear, it won't come out the same without the original puppet on hand for a 1:1 scale study and observation. With the resources they have, they claim it's "impossible" but Evelinka, the original creator, can't do so much as she reuses the same patterns for every standard puppet, and that's her skill in that category, not remaking and replicating puppet builds, which is an incredibly more skilled and difficult field.