Maybe to start there could be mandatory disclosure of all donations/vacations/other lobbying tactics that each senator is engaged in, correlated with their track record of voting and publicly disclosed and actively monitored.
I agree it's a hard problem, but I think with a big enough pool of information everyone could get a letter/email saying that politician x cost you y amount this year by voting this way. I know I would be compelled to make some changes.
1. Promise to give lots of money to the senators who vote for their interests well enough. This doesn't even have to be targeted.
2. A neutral party collates all that information, and publishes it.
3. Wait until the first sufficiently-helpful senator retires, and is no longer bound by the rules.
4. Reward them.
5. Wait.
6. Profit.
Better than the status quo, since only huge companies will still be able to effectively lobby, but introduces problems of its own. That is, assuming the system works as intended. Which it won't.
Because how do you make sure the disclosure is mandatory? This punishes honest lobbyers (and honest lobbyees) more than it punishes dishonest ones; if all politicians are being lobbied evenly from all possible causes, but cause B's lobbying is secretive, politicians are incentivised to vote for cause B (i.e. against ¬B, which they are being publicly lobbied by).
This system does incentivise not being the ones to lobby, though – assuming there's no way to do lobbying that appears to be ¬B lobbying, so the politicians can be virtuous and vote B, which makes the “¬B lobbyists” lobby harder because obviously their bribes weren't big enough…
I agree it's a hard problem, but I think with a big enough pool of information everyone could get a letter/email saying that politician x cost you y amount this year by voting this way. I know I would be compelled to make some changes.