You're making this way too hard. You don't need to light the backdrop perfectly evenly. If the dimmest area of the backdrop is blown out the rest of the backdrop will be equally blown out. There is no shade of white brighter than blown out.

Here's the problem with using an umbrella to diffuse the light or moving the light further back: You're lowering the amount of the flash's output that makes it onto each square inch of the background. That's going to make the backdrop dimmer overall. This means you're more likely to get varying brightness levels from different parts of the backdrop.

The key to getting uniform white backgrounds isn't illuminating them evenly. It is making sure the dimmest part of the white background is lit bright enough to be pure white with the ISO and aperture you are using. (Shutter time matters also for constant lights but not for strobes since the strobe's duration is usually shorter than the camera's sync speed.)

It doesn't matter if one side of the backdrop is getting more light than the other. You just need to expose in such a way that the dimmest part of the backdrop is completely blown out. You'll find this easiest to do if the flash lighting the backdrop is at full power (assuming it isn't too bright as to create a lot of spill back onto your subject).

If the dimmest part of the backdrop is pure white in your resulting photo the entire backdrop will be pure white. There's no way for the brighter parts of the backdrop to be brighter or whiter than pure white in the resulting image.

Once you've got the exposure right for the backdrop you can then adjust the power of the other light(s) to properly expose your subject at the same ISO and aperture.