this is a intersting scenario for an attack of the network.
Assuming some big actors (financial institution and governament of the West, in this case) decide to hard fork #Bitcoin in BetterBTC, will be initially for a blockchain where allow only transaction compliant with money laundering law. That will require to have miners on their side. Miners care about profit, and illegal transactions on BTC will probably have higher than normal fee for counterattack BetterBTC for keep miners on their side.
Western institutions will not propose immmediatly Proof of Stake changes because thats will push away Miners (they strenghts is the hardware, which doesn't matter in Proof of Stake)
Proof of Stake will also decrease network security, since most of actors know that with enough founds, you can control the network
For Western instituion there is also a bet to make: if they move large capital of BetterBTC, and BetterBTC will be less secure than #Bitcoin, they will risk to lose everything. Remember that BetterBTC will still be a cryptocurrency which can be attacked from bad actors
Other nations not aligned with the West might also decide to ignore BetterBTC and still use BTC
Of course, this is just my opinion of what might happen