I like the idea of the play money (something you can see, while you discuss, as opposed to debating points you have to keep in memory), but you need to take it past the first step you’ve described.
Based on the description of the scenario, one of the challenges is getting prioritized citizen buy-in for all those things they think are reasonable but not necessarily their personal priority. So maybe you need to give them 3 denominations of money ($10, $5, and $2?) and get them to assign it how they see fit. Some folks will undoubtedly put all their chips/money on ONE thing, but others may realize that, while X is important to them personally, Y actually requires more funding to be accomplished. They can put $5 on X, $10 on Y, and $2 on something else, or $7 on X and $10 on Y, or whatever.
The objective here is to bring them into the same mindset you folks have to adopt; which is a fixed pot of money that has to be chopped up and accomplish a lot.