Exposure compensation is just another means of changing either shutter or aperture. I is not some fourth component of exposure, there are still only three: ISO, Shutter Speed, Aperture. If you have manually selected an aperture, changing EC will reduce shutter speed. If you have manually selected a shutter speed, changing EV will increase the size of the aperture, unless it's already at max.
ISO 300 is NOTHING. Cameras these days can easily be used at ISO 1600 or even 3200 without serious problems with noise. I think your being much too critical of noise, and I think your severely hurting yourself by forcing the use of ultra wide apertures or ultra slow shutter speeds, both of which can affect IQ in ways that cannot really be fixed in post. Noise is predictable, and therefor easy to reduce and clean up in post. A thin depth of field, however, cannot be fixed in post...if you end up with too much of your scene out of focus because you used the maximum aperture, your simply stuck with too much of your scene out of focus. If you end up with motion blur or camera shake blur, then your image is simply blurry.
When it comes to deciding which tradeoff to make, you should generally make the tradeoff of more noise first, then move to a wider aperture or a slower shutter speed as a last resort. I would also make sure you fully understand what exposure compensation is and what it's doing. EC isn't some fourth option for changing exposure...it is simply an alternative means of changing shutter speed or aperture. If you select maximum aperture and an ISO setting, then using EC to increase exposure means your exposure time (shutter speed) is getting longer. That means you run an increased risk of blurring the scene because of shaky hands or subject motion.
Noise, as much as it is demonized these days by photographers, is one of the more manageable artifacts in digital images. Don't fear it...at least, not as much as you do. ISO 300 is VERY, VERY LOW. You should be pushing ISO 800 or 1600 before you really start worrying about noise. With a good camera (modern APS-C DSLR at least), you should be able to use ISO 1600 without fear, and FF DSLRs like the 6D can be used at ISO 6400 or higher. For compacts, many use very advanced processing for JPEGS, making ISO 1600, 3200, and sometimes even higher very viable options.