Don't use QT, use the native H.264 encoder that comes with AE/ AME. Regardless, getting 10 MB files at those resolutions and expecting pristine quality seems like a dumb requirement. It would be tricky even with a normal HD 720, so for a triple full HD 1080 res it's illusory, considering that depending on settings a single frame can have anywhere from 1 MB to 30 MB in the most common use cases. I can't help but feel that your client is not understanding something fundamental here.
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It looks like you need to do some studying on compression techniques. If you set the quality to 100 the files are going to be bigger than if you set the quality to 50 or 75. Are you positive about the dimensions? 3840 X 1080 is a non standard size. QuickTime does not handle H.264 well, is not supported or being developed by Apple any more, and is not recommended. The Render Cue and Output module will not do multi pass compression which is required for the highest quality.
I would start by verifying the non standard frame size. H.264/MPEG-4 AVC - Wikipedia
Use something like HandBreak if your client insists on that frame size but be aware and make your client aware that they are swimming in uncharted waters and things may go wrong. I'm with Mylenium, that file size for that frame size and that program length is going to be very hard to do unless you drop the frame rate, and there are standards for frame rate also.