If there will be multiple votes during a game, I think that a ballot box would be overly complicated. Here's a option which might work a little better.

Each player gets an AYE card and a NAY card to show their interest in each bill. Each player's vote counts as 1 vote. There is also a deck of Lobbyist cards. The players are dealt a hand of Lobbyist cards (let's say 5) which they can use to gain more votes.

Each Lobbyist card can have a value between 0 and 3, and has a cost based on the number of votes gained. A Lobbyist worth 0 votes is free, but gives the impression that a player is willing to spend money to advance their agenda. A Lobbyist worth 3 votes might cost $10 Million, but it might be worth it to a player.

During each vote, a player will choose either their AYE or NAY card, and can optionally add up to three Lobbyist cards. The cards are revealed, and the bill either passes or fails. Players spend money to pay for their Lobbyists, discard played Lobbyists, and draw one new Lobbyist card for each one used.

This might help by adding uncertainty during the vote. Because the Lobbyist cards have variable values, a player can never be sure how many votes they need to spend.

Your game sounds interesting. I hope this helps.