Is the act of recording the usurious interest amount received from a non-Islamic bank in the accounting ledgers of the hospital where one works considered a sin and usury?

It is not permissible to open an account in a usurious bank except out of necessity. It is also not permissible to open a savings account if a current account can be opened. However, if only a savings account can be opened, then there is no harm, but the interest must be disposed of by spending it on public welfare. If the hospital owners are not compelled to open a usurious account, or if they benefit from the interest, then it is not permissible for you to work with them, as this would be assisting in sin. But if they are compelled and spend the interest on public welfare, such as treating needy patients, then there is no harm in working with them.