Just me, but probably not.
LED continuous lighting, particularly if we're talking little panels, tend to have relatively low light output and would be insufficient to match sunlight no matter what camera/lens combination you were using.
f/1.4 is nice, but you can buy f/1.4 for MFT, and moreover aperture affects any light source. While it might help brighten up your subject lit by LEDs, it'll also brighten up the sunlight on the background at the same time and you'll likely overexpose. And the sensor size increase between an APS-C camera vs. a four-thirds camera isn't going to be giving you that much more than a stop of added dynamic range, and you still can't equal or overpower bright sunlight with a weak source of illumination.
My suggestion would be to look into lighting with multiple sources, and shutting the curtains :) or rethinking your light setup.
Overpowering sunlight is hard and often requires a really big powerful light. If you take the sunlight out of the equation, you can balance your background light with your key much more easily, vs. using the sun as your background light.
If you use the sun as your key, however, that could be another way of doing it. Put up a scrim (sheer curtains, paper) in the window to get diffused sunlight, and it might not be a bad way to light your subject, and use the LEDs on the background or as fill, so they don't have to match or overpower the sunlight.
See also: dpreview's youtube video: "Take better indoor portraits with natural light (without buying any new gear)".