Section 4 Summary

Get out the microscope, because we’re going through this poem line-by-line.

In the swamp in secluded recesses,
A shy and hidden bird is warbling a song.

Not only do we have another mood change here, but we also have a change in setting. Now we're in a secluded swamp with a hidden bird singing a song, and a frog playing the banjo—oh wait. Still, the mood is a bit strange, mysterious perhaps with those "secluded recesses."
Even the language here is different, without all the flowery adjectives, which adds to the sudden mysterious mood we have here.
So near that swamp in "secluded recesses," a shy bird is "warbling a song." Songbirds also tend to become symbols in poetry, usually for our inner "song" or soul. So they might mean the same thing in this poem later on. Keep an eye out for that.
But what we do know for sure is that the imagery here is strange, in a dreamlike sort of way, since we don't usually imagine birds singing in swamps (Kermits only). So we have even more reason to suspect that the bird may symbolize a more mysterious world, part of the stuff we can't always see or understand.
That would makes sense, since when we try to rationalize death, we often end up grappling with this mysterious and unknown world.

Solitary the thrush,
The hermit withdrawn to himself, avoiding the settlements,
Sings by himself a song.

Hmm—more weird stuff here. In addition to being a songbird, we have a thrush who's a hermit "avoiding the settlements."
The bird "sings by himself a song," isolated in the woods somewhere.
The mood has become even more isolated and mysterious at this point, with the hermit-bird singing to himself. There's no one around to listen, so you know this guy isn't looking to be on American Idol. Instead, it looks as if he is part of this more soulful and unconscious world that belongs to the individual alone. It's his world and his song, no one else's.
Since he avoids the settlements, we know this life is of his own choosing. No one is forcing him to sing. He's doing these things for his own purposes without looking for anyone's approval, understanding, or company.
If we put this in the context of grieving, the weirdness kind of makes sense. When we grieve, we tend to grieve alone and we feel things in our own way. So this hermit-thrush who "sings by himself a song" seems to represent the more unconscious world that's associated with grief. In other words, it's the world that can't be consoled with fruitcake and flowers. (Try us, though. We're partial to long-stemmed roses.)

Song of the bleeding throat,
Death's outlet song of life, (for well dear brother I know,
If thou wast not granted to sing thou would'st surely die.)

Man, it looks like that hermit-bird has been singing his song for an awfully long time, since he has a "bleeding throat." This tells us there's quite a bit of feeling behind that song since he can't stop singing it.
Line 23 also gives us some figurative language with that "bleeding throat," since we can assume his throat isn't actually bleeding. That would be one tough bird.
Line 24 has some more blending of life with death in the idea of "Death's outlet song of life." That song is an outlet for the bird's life, keeping death away. In this sense "Death's outlet" means something more along the lines of "escape from Death."
The little parenthetical clause (the part in parentheses, kind of like this part of this sentence) that we have between lines 24 and 25 also gives us a sense of the speaker's empathy with the thrush, which is referred to as a "dear brother."
The speaker says that, without that song and the bird's ability to sing, life would cease to have any meaning. Instead, he would "surely die" without that ability to express the song he has within him.
On a deeper level then, that little message gives the suggestion that expression and song are necessary to life. We, like the thrush, cannot live without "song" and the ability to express our emotions, especially when we're suffering pain. That expression of the soul through song is what helps get us through all the pain.