How does one demonstrate compassion without the pity getting you down personally?
By accepting the fact that this suffering is just the nature of Samsara? And that all beings who are stuck in Samsara will suffer and nothing can be done about that? Except to show them the way out of samsara? But the only way to do that is to find the way out yourself. Because obviously you can't show another the way out if you don't know it yourself. Enlightenment is the only way
out of this suffering so we must get this enlightenment because it's the only thing
that will actually help them. Without that, you can't actually do anything that will really and substantially help any of them.
But of course, if one of them needs food and you have some food, it's helpful to you and them to help them and give them some food so they don't starve to death. But obviously, some will still starve because you just can't give food to every being that needs it. That is impossible. Just to accept the fact that this is impossible, goes a long way to not getting you down personally, IMO.
This is Samsara. Yea, it sucks. It's completely filled with suffering. But that is why it's called samsara to begin with. Lamenting over the fact that it sucks is really not all that productive, it's counter productive really. More important is what to do about it.