I say go for the birdcage. Your dress has a lot going on at the bottom and I think the long veil actually gets lost in all the detail. A bird cage really pops against the simplicity of the top of your dress, and brings the eye up to your face too.
I've actually seen TONS of brides with "complicated" dresses on this site. And not a single one of them looked out of place, they only looked beautiful, radiant, and confident. I will be getting married in a dress that I never thought I'd wear myself. I envisioned a simple, flow-ey, light dress. But when I tried on my puffy A-line gown I fell in love, and I have not regretted my decision a single time! Wear what makes you feel good and nobody will think twice about the bride looking "out of place"!