A few things to consider here.
1) You change your project to 32bpc (bits per channel) but that does not mean you are rendering a 32bpc image. Pretty much every way of rendering out of AE gives you an 8pbc file. Whilst processing in 16 and 32 is pretty common now, the file sizes would be so huge that our output is generally 8 or sometime 10bpc depending on format.
2) The confusing thing here is that you should be rendering a 32bit file, just not a 32bit per channel file. A 32bit file has 8 bits for each of the 4 channels; Red, Green, Blue and Alpha. In AE this comes up as selecting RGB+A in the render dialogue. The 8bpc image you render will still process in 32bpc if that is what you selected for the project and produce the effect you get in that mode. The 4th channel (the Alpha) will tell Avid which parts of the image to make transparent and which to make opaque.
3) Avid expects straight (or unmatted) alpha channels. Many programmes premultiply the RGB channels by the A channel as a way of saving space but make sure you change this in AE on render so that Avid will interpret your Alpha correctly. Blurred edges and glows in particular will behave strangely otherwise. The other thing to consider is that you will have to invert the alpha channel on import to Avid in order for it to be interpreted correctly. This will be selected by default most likely, but just double check.
Andy