Sorry that am seeing this late. But I think I have a solution which may help others. What you've described is complete to pick a member with simple random sampling (SRS). To select other members, they're statistically many other ways - but the easiest is to base on the selected member and then select for instance every 2nd in order to add the other 2 members and have the list of three
I can see that you had generated a list to pick from, say if the random member is one in position 2; then additionally pick 4 and 6 in case its every 2nd; or pick 4 and 7 in case its every 3rd. That way, the one you pick first provides the random start and thus every other systematic pick remains at random in the list.
To achieve this in the form, include calculate fields that contain the names of members depending on their position - each member a variable. Ounce its the selected position, then call the respective member.
Hope it helps