Actually the answer whether a 1.4 tc will work is already in the image.
TS posted an image 480x320.
The diameter of the illuminated circle, coincidentally, is ~480 pixels (width of the image).
However looking closer, the diameter is actually < 480, so we use an estimate of 420 pixels to be on the safe side.
The diagonal of the image is 577 pixels.
With a 1.4 tc, the image scale will increase by 1.4x. The diameter of the illuminated circle will likewise increase by 1.4x. Hence 420x1.4 = 588 pixels.
The enlarged illuminated circle must cover the image diagonal, which is 577 pixels. Since the enlarged illuminated circle is 588 pixels, it does cover the image diagonal.
So short answer: yes the 1.4 tc should work, or at least drastically reduce vignetting. However the margin is pretty narrow, TS may want to try out at the shop before committing to a purchase.