Is spending on a sick and temporarily impoverished brother—despite his possessing funds he doesn't want to spend to secure his daughter's future—considered charity or not?

There is no objection to spending money on a brother's treatment with the intention of charity, and the doer will be rewarded for it, because voluntary charity is permissible for the rich, unlike obligatory charity like Zakat. It is, however, recommended for the rich to refrain from accepting voluntary charity. Anyone who is forbidden from receiving obligatory charity may be given voluntary charity. It has been narrated from the Prophet, peace and blessings be upon him, that a Muslim's expenditure on his family is an act of charity.