If you decide to start a new proof-of-work chain for BTCv2, then all I have to do is get all the old BTC proofs-of-work, and add them to the new chain, and I'll instantly have lots of BTCv2s to my credit.
Problematic?
Thankfully, no. The Bitcoin mechanism has the concept of a genesis block hash, which is the theoretical hash (0x000000000019d6689c085ae165831e934ff763ae46a2a6c172b3f1b60a8ce26f) of a nonexistant 'root' block. To start a separate block chain you'd start with a new genesis block hash and start adding new blocks to that chain rooted at your new genesis block. Since each new block contains a hash of the previous block in the chain, you could not re-use blocks for the current block chain in your new block chain.