They don't need to invest much in Bitcoin. They issue loans nominally in Bitcoin and hand out checkbooks. They don't have to hold the full amount of the checking accounts in Bitcoins, just a fractional reserve. And their loans get paid back in real Bitcoins.
Pretty soon they have a lot of Bitcoins.
Yes it can happen that payments with banks will be bitcoin but I don't think that they will allow that bitcoin will replace fiat for real. As we know that the risk is still there and because of the price of bitcoin is not stable enough and cannot be controlled. I don't think they will completely trust their business to a decentralized crypto's.
Bitcoin as a high value store token has been gaining popularity among most of the financial service providers. Banks were the most among those and banks supporting bitcoin won't happen as quoted. Already banking system is with continued plan of updating themselves with the technology but haven't issued willingness to use bitcoin into the banking services.