Oh, lots of great feedback! This is exactly the kind of stuff I'm looking for. I'll try to address the points I can.
Evileye: Did you see the sketches I posted for the floating HUD? What do you think of them?
Helm: I looked at a bunch of Lionheart stuff, and while it has minimal HUD space, it definitely uses depth to the platforms. The character himself is rendered in flat profile, though.
http://www.mobygames.com/game/amiga/lionheart/screenshots/gameShotId,231762/I would consider dropping his left foot down a pixel to give the illusion of depth. I don't think this minor change would look too horrible when flipped. Your take on the sprite rendering style is interesting -- the purple outlines for where the light hits the sprite and the extra shades within the colored regions (while the shadows remain black). It makes the sprite look less flat, but I'm not immediately sure that's the direction I want to go.
As for the HUD, I have no problem axing the cheesecake. Faulkner said, "Kill your darlings", and it's as applicable with pixel art as it is with writing.
Those Alisia Dragoon shots are interesting food for thought, since they use flat profile tiles as well as tiles with varying depth in different parts of the game. I would want to keep mine consistent to make it easier to build levels.
Vierbit: You have a good point about making organic tiles. The trick for me would be building those shapes into a tileset without too many excess tiles. I'll look into it.
Buloght: Do you think if the bird matched the detail level of the character sprite it would still look out of place?
Ben2theEdge: Good call on the asymmetry. If I go with a big HUD, I'd have to mask off a section of the field of play -- a rectangle that is only used for the HUD, and has no transparency (so it wouldn't interfere with playability like it does currently). It would provide a level of awesome, and that's more of my focus. I am concerned with the complaints of style clashing. I'm not interested in making a huge HUD that matches the comic-like style of the sprites. If it's gonna be big, I want it ornate and fiddly like the HUDs I did before. Otherwise, I could make a free-floating HUD like the top one in my sketch and render it in the same style as the sprite.
Philipptr: You're right, adding tiles to make the edges of the platforms less rectangular is an easy fix. As the mockup is right now, I think there are bigger problems I need to address first.