You might check the help file for... watermark
I'm not sure of a way to have the logo appear on your videos without adding it during editing.
Unless you're just superimposing it electronically as you're broadcasting -- the way networks do.
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The file size will not be increased, regardless of how many logos you add (or how many Video Tracks you have), as you still have the very same Frame Size, the same Duration and will be using the same Bit-Rate - exactly the same file sizes.
For logos, I create these in Adobe Illustrator as Vector Art, and then Open them in Photoshop, where I Scale and Position as is needed with a Transparent Background, and Save_As a PSD, which will Rasterize the logo. That PSD is then Imported, and used on an upper Video Track, for the Duration that I require. Work very well.