Is it permissible to accept sponsorship from an interest-based bank to cover the costs of purposeful community activities, and is it permissible to participate in these activities if the sponsorship is not permissible?

If the bank's gift is not conditional upon promoting it, then it is permissible to accept it, though it is disliked due to mixing with its money. However, if its undertaking of activity costs is in exchange for promoting it, then this is not permissible, as it involves cooperation in sin and aggression. Participation in the activities is permissible as long as it does not include promoting the bank's forbidden dealings.