I don't think it will cause Bitcoin's crash. People who are heavily investing in BTCs aren't common people, they are rich people, they know what they are doing, it banks were a problem for them they wouldn't be putting so much money into Crypto-Currency world.
And there is the possibility people inside the banks invest in BTCs and make profit with Crypto-Currency too, not possible to prove, but it's possible. Bitcoin price won't crash, but the difficulty of using it can increase for the mass population.
I agree with you, but there are options to short BTC too, therefore profit with BTC is possible for banks in every case.
The institutions which will start to fight BTC and cryptocurrencies are governments because they lose control over the regional money flow (money laundering, taxing,...)