Good idea!
What's wrong with swapping previous race standings for the start of the next race? You could change the points around even more so the top 15 get points, then you'd have some drivers slowing down so they're at the front for the next race, some just finishing in the middle every race and hoping to even out a win, but the best would always be fighting from the back to get to the front.
Also, what's wrong with removing some of the rule limitations, add some variability into it again and make it the best of everything once more. Teams with different engines and adjustable wings and stuff would add a massive variation in performance between teams, even if you end up with teams that are being lapped 5 times it'd be much more exciting to watch!