my father used to be very well off, and had countless meshulachim a week, but he would give what he felt he could, and what he felt befitted that cause (even a rich person can only give $1000 or $5000 once in a while to exceptional causes) and if someone said (especially chutzpadik-ly) that $360 wasn’t enough he made the meshulach give it back and told him to leave. if you give $360 to thousands of organizations it really adds up, and the halacha is to give to more people, not all tzedaka in one place. If someone wants to be brazen they don’t deserve it.
At the same time, you can feel out the person, if you ask in a kovodika way if they can handle any more as the organization really needs it, that might be ok, but when you say it leave an “out” like “I know times are tough financially now, if you can’t I understand & I’m grateful for what you did give, but our organization is really in dire straits, can you possibly afford to help out a bit more?” then perhaps you have a chance, but again, feel out the person. In general mentchlichkeit will get you much further than brazenness and chutzpah.