Short answer: not possible. Less file size = less quality. Plain and simple.
Long Answer: it is possible to get acceptable quality without the file size being too big. What's Savin you is the resolution being small. If you were working with 1080 files under thing under 8mbps would look pretty awful. However, at that resolution, 700kb - 1mb should be fine.
As mentioned, something like MPEG streamclip could do the job in the simplest way possible. In my experience though I like to export straight out again rather than compressing the file. What settings are you using on export? Mainly bitrate and codec. This will help in deciding if it's better to compress a file you have or if it's better to export with different settings.
P.S the file size not changing when you change the resolution actually makes perfect sense, just doesn't seem like it should. The bitrate is likely the same on both (each second of video has, for example, 1mb of data) so the only effect resolution will have is how many pieces that data has to fill. Fewer pixels = better quality pixels. The file size will only change if the bitrate changes.