I'd have to back up what has already been said. For best results, you will need to export as a Quicktime Animation using RGB+Alpha. Yes, the file size will be huge.
However, there are things you can do to reduce the overall file size, and these things have to do with how your project is set up and your workflow.
Make your Composition size the size of the lower third, not the size of the canvas size for the end user. If the video being produced is 1080, the lower third will be at max 1920 wide, but will likely only use a slice of the vertical space. So size the comp to that and let the editor place it where it needs to go.
Secondly, render the lower third without the name, have them bring up the name in PPro, now he will only need 1 file.
Having not seen your lower third, I can only assume it animates on and off. Does it continue to have motion while up? If not, render only the build on, and build off. And render 1 frame of the static. Let the editor use the build on, then frame hold the static section, then build off files.
Last, if your graphic does have consistent motion, you can either animate in a way that loops to reduce the length and allow the editor to repeat the loop. Also, and I have done this, is- break the composition down into multiple outputs of animation files which can be different sizes, or loops. Things like glimmers, flares, etc, they may only be a short duration of the overall graphic animation. You can isolate those out and render as separate files.
Yes, your editor will have to layer. But PPro can handle that. He can create a sequence for the lower third, and use it as a nested sequence inside his main sequence to apply the names.
My suggestion is think hard about workflow and output possibilities, and go with, as others have said, QT Animation with Alpha. Anything else will give you sub par results.