I agree you don’t want bonded contact between the face and the cushion. 

Transient isn’t going to give you a more realistic result than Static Structural. Transient is useful when the inertia forces (mass x acceleration) are significant compared with elastic forces (stiffness x deformation). Transient would allow the solution to begin in the case where there are only applied forces and contact forces on the mask. The solver will compute the inertia forces to put the mask in dynamic equilibrium.Imagine having the mask 1 mm off the face and letting go at t=0. It can accelerate and slap into the face.  You can’t do that in Static Structural. If all you have is forces and contact, you need the contact to be frictional and to be initally penetrating in enough places so that there is a static equilibrium in all 6 degrees of freedom (3 translations, 3 rotations), otherwise Static Structural will give an error at the beginning. 

Adding supports such as Remote Displacement in Step 1 of a Static Structural can overcome the problem of getting started. The Remote Displacement can move the mask closer to the face allowing the cushion to build up contact with the face. In Step 2, the Remote Displacement can be Deactiviated so that only forces and frictional contact forces are holding the mask against the face.

Static Structural models with initial penetration in the frictional contact can often resolve. If the solution fails to converge, you can use Part Transformation to move the mask (or face) to a position where the contact has a smaller penetration.