SOOW is 600 volt cable and much fatter than say SJOOW. THE J stands for junior grade and is rated 300 volts. More than likely it will not fit. The increase in size is all in the extra insulation.
I'm a bit late to this. Google says the device is 7.5 amps. Lets aSSume 16/2 cord.
My quick reference shows 16/2 SJOOW at 0.320" (8.13mm) and 16/2 SOOW at 0.370 (9.4mm). 16/2 SJEOW is usually slightly thinner than SJOOW, while SEOOW is about the same as SOOW. I suspect the internal cable clamp could fit any of these because they usually give lots of room to make assembly easier.
In general I much prefer SJOOW and SOOW over SJEOW/SEOOW (and yes, I know SOOW is "only" rated to -40, is slightly heavier and has inferior abrasion resistance). Any of these would be more than adequate, but SEOOW is probably the least forgiving if the cord grip is tight.
For the strain relief I usually just layer some heat shrink. If you order some adhesive lined heat shrink it should be easy to build up something to fit in the handle. I would probably not use adhesive lined heat shrink for the strain relief itself on SJEW or even SJOOW because it will be too stiff in the cold, relative to the cord, and doesn't let the cord move within the strain relief (that's why your factory cord failed in the first place - too much strain and not enough relief!).
Another option is to make your own strain relief from larger flexible cord. I know from experience that the outer jacket (insulation) from 14/3 SJEW will fit over 16/3 SJEW. I've mixed and matched many other sizes as well. If you're stuck, some super 33+ e-tape around your home brew strain relief would probably give something for the plastic handle to 'bite' into as well.
It's hard to see the connectors on the switch, but you'll either have to splice or replicate the factory connection.
Hope that helps...