[GolfPopper]

It's been a long time since my astrophysics, but I think the seeming contradiction you're running into might be from treating lux (illuminance) as a measure of emitted energy, when its actually a measure of received energy.

The Sun's (or any star's) emitted energy is measured in terms of solar luminosity.[1] The nominal value of solar luminosity is 3.83×10^26 watts. At twenty five times as luminous, Sirus' luminosity is 9.5710^27 watts. We can divide that by your 296 billon times, which gives.. 3.2x10^16 watts as what actually makes it to Earth. If the we convert that back into solar luminosity (to figure out the apparent brightness at Earth), its 8.3595 10^-11.

Now, if we look up at the sky, and check how bright the Sun and Sirius are from Earth on the magnitude scale, which each step is ~2.5 times brighter than the one below it (and vice versa), the Sun has an apparent magnitude of -27, while Sirus' is -1.46. I.e. the Sun in the sky is about 8 billion times brighter that Sirus is. That's within an order of magnitude of what its calculated solar luminosity should be. Again, it seems about right.

1. https://en.wikipedia.org/wiki/Solar_luminosity