"picking the 4,000 participants and deciding who will get basic income and who will be part of the control group, which will not receive additional government funds"

You do not "pick" the people to be part of one group or another. For the study to be scientific, You do a random draw. You could subdivide by categories of interest (e.g. age, employment status, family status, income, (dis)bility), then run a random draw within each combination of such categories. At the extreme end, every participant is matched with a "twin" with similar characteristics, and one "twin" is randomly selected for BI or control.