a. won't cause my bank to close my account
Even how reputable the crypto exchange is, if your bank have a self policy about cryptos then we aren't sure here about closing the account issues. Like here in our country, there's a bank who closes account of their clients because of continous withdrawal from a crypto exchange. Im not just sure if deposits is involved.
b. won't take days to complete the transaction
At most of the cases it really takes days.
Generally, if you google some exchanges that accepts bank deposits and withdrawals, all are reputable. It's just that we doesn't have an exact positive reviews since anyone have different experiences. Country and location is also a factor. May we know from what country you from? Try to explore also if there's a local exchanger there as in most of the cases, they are accepting bank transfers.