I like your idea. Playoff teams of course end up with picks 13-20 while the other 12 are a complete lottery. This would work for both the FA and rookie draft.

I'd still weight the lottery, but instead of the normal:
12,11,10,9,8,7,6,5,4,3,2,1 for 78 chances(15% chance all the way down to 1.3% chance)

I would switch it to

14,13,12,11,10,9,8,7,6,5,4,3 for 102 chances(13.7 all the way down to 2.9% chance).

Thus worst record still has the best chance at the pick, but now to a much lessor degree.