I also use my cards extensively for paying bills. The BMI mastercard is worth considering, but so is the Tesco card, and that now lets you earn Virgin miles, so going forward once the BMI card becomes Avios earning, the Tesco card will give you either Avios or Virgin miles, depending on your needs and preference.
You won't find the benefits of the Barclycard... as there aren't any worth having! Cash back can cause problems if your self employed paying suppliers, as it is probably taxable, whilst miles with no cash value aren't!