If there is was a bitcoin bank, you will kill online wallet for sure , bank will still stand like it was before with its own function, that's why online wallet do prepare to exchange your bitcoin to real money, because online wallet knows that bank accept money not bitcoin. and any fee also for their income. not start a war, because still many people using money than bitcoin, and many people exchange their bitcoin to money even not directly from bitcoin to money. (bitcoin - exchange money through wallet - cashout from bank the ending is in money form ). CMIIW
They both cannot get into war because banks are getting benefit from bitcoin because people are cashing out bitcoin through banks, both are connected to each other