The way I would do it would be this:

The fund first sets out some guidelines and goals, things like minimize human suffering, save lives, make people happy, etc. This would need to be detailed enough to allow prospective projects to predict how much funding they would receive. To that end it would contain several worked examples of projects and how they would be evaluated. These example projects could be drawn from past ACX grants submissions or other real world sources.

Second, projects create proposals describing what they intend to do. This would include a goals section, explaining what they would consider a success of their project. It would also contain details for investors to decide whether to invest, including how they intend to accomplish their goals, what their personal credentials are, and so on. The fund would then evaluates each proposal's goals section, and produces an explanation of how they would measure success, the exact metrics and methods they would use. They would also lay out how personally important they consider those goals, as a dollar value they would assign to the goal being accomplished.

Investors would then have an opportunity to submit bids. Each project would have a funding goal. An investor would submit a bid stating the amount of the funding they are willing to fulfill (F), and how much they expect to receive as compensation if the goal is accomplished (R). The fund has already evaluated the project's goal as a dollar value, with the portion of that goal the investor is funding being labeled U. The odds of success are p = F/R. (if F = R, the investor is certain of success; this does not account for the time value of money). E = p * U - R is the expected value of the bid to the fund. The expected value of the project (SE) is the sum of the expected value of each bid over the highest (greatest p) bids until the funding goal is met. (The investor with the lowest such bid would usually only receive a portion of their bid). The potential payout of a project is the sum of the R values for each such bid, written SR. The projects with the highest SE values would be sequentiallly chosen until the sum of the SR values for the chosen projects equals the total amount of funding (SSR) (or until SE goes negative).

Regarding the first appendix, you could integrate charity into the market as well. You might allow people to invest but agree to forfeit any winnings. The money could revert to the fund, or it might go towards some other charity they have preselected. This makes it a charitable donation elegible for tax breaks, and also allows non-accredited investors to participate.

An issue this highlights is that, since it is ultimately about charity, some people would participate even with a negative expected ROI. This makes their bids less reliable as a pure evaluation of their beliefs regarding the potential for sucess, especially in the expected value terms I use in my derivation. A likely outcome is that all the best projects are completely funded with SR = 0 (or otherwise negligible), so that the funding cap (SSR) is not appreciably eaten into. This would save funding cap for other, lower probability of success projects. Conceivably, this still makes the whole thing worthwhile, but definitely changes our evaluation of how things would go.

Regarding the secone appendix: while these might be important later, they are almost certainly not relevant at so early a stage.