I think the purpose of this question is the look at the balance of forces and moments, which must be the same for the 3-force group and the 4-force group. If the 4-force group is supposed to be support forces, which isn't clear to me, multiply the results by -1.
Assume disc radius = 1. It could be any radius, but it will divide out and cancel when looking at the moments, and doesn't matter for the forces, so just use 1 for simplicity.
(eq1) net force
(F_C + F_L + F_R) = (F_NW + F_NE + F_SE + F_SW)
(eq2) moment about North-South axis (let east-side-down be positive)
F_R.sin(120) - F_L.sin(120) = F_NE.sin(45) + F_SE.sin(45) - F_NW.sin(45) - F_SW.sin(45)
(eq3) moment about East-West axis (let north-side-down be positive)
F_C - F_R.sin(30) - F_L.sin(30) = F_NE.sin(45) + F_NW.sin(45) - F_SE.sin(45) - F_SW.sin(45)
Note there are fewer equations than unknowns (the wobbly-table problem)! I think this means you could set one of the 4 to zero for simplicity. (but the direction of the opposite one may flip as a result). Alternatively, have the two pairs of opposites differ from the average of all 4 by the same amount (with opposite signs).
... hopefully that gets you going, bonus points if I screwed it up!