I dont know how a bitcoin bank would work but a bitcoin insurance company might be something to pursue.The idea is that premiums are worked out based on the risk of losing wallets and the rate of fraud with paypal chargebacks and other calamities that could cause a loss of your wealth.You would pay a premium each month and cover a set rate of any losses you suffer.
Would you pay an insurance premium for such a service with bitcoins and what measures would the insurance company need to take to protect its members from fraudulent claims ?
I don't know the reasons why you asked this question but the whole idea seems stillborn to me, just like the one about banks providing loans in crypro (frankly, it is even less feasible). It looks kinda impossible given bitcoin's anonymous nature, so I really doubt anyone would seriously consider running such a business unless they are in fact pursuing what they should take measures against, that is fraudulent claims and schemes.
The only problem is, how would the insurance company verify if you indeed lose your wallet?
How could they see if the amount you take the insurance for is really the amount you own?
I think you would have to use a multisignature wallet, if such exist. But then it would have very little to do with insurance.