I wonder if we could get lawmakers to have all stock transactions to go through a fund that they directly (or if they choose, indirectly) run. This would enable them to buy individual stocks, similar to what the do today, but with the major exception being that others could buy into those funds and benefit in the same way. While it wouldn't address the corruption, it would make it so that they aren't the only ones benefiting from it, and would limit the upside since many others would have a similar position in the stock.
I'm sure there are many reasons why this is worse than the current situation, but I can't see any at the moment.