Generate batches of random numbers that each are zero-sum and for which MAE is uniformly distributed over specified interval

In the context of polls of voting results, i want to generate random numbers with specific properties to sample the possibilities within the error of margin.

For example, suppose i have polling results:

• Party A: 34%
• Party B: 25%
• Party C: 16%
• Party D: 26% With a margin of error of 3.2%

I want to generate batches of 4 (because in this case there are 4 parties) random numbers. Obviously because voting is a zero-sum game, the numbers in the batch need to sum up to zero.

Given the error of margin, i want each element of the batch to be (in absolute value) less than that.

An example could be: (-0.5, +1.2, +0.1, -0.8)

When generating a large amount of these kind of batches of random numbers, i would like these to have some specific statistic properties:

1. The maximum of the absolute value should be uniformly distributed over (0, error_of_margin). (This is the easy part)
2. The mean of the absolute values should also be uniformly distributed over (0, error_of_margin)

I tried two different approaches. I will link a gist, to not fill the question with code. https://gist.github.com/thomvil/02890ea2873eed6bb155e7dc387c9564

Does anybody have some suggestions on how i could tackle this?

Thank you