Piles and pile groups exhibit nonlinear load-deflection behavior, under both axial and lateral loading. Unless you can closely control either the loads or the movements, there will be no "stiffness matrix" or constant stiffness.
In any event, the stiffness will be a function of the load or deflection, and will be "constant" over only a small range of loads. Don't let elastic solutions fool you - try varying the stiffness by factors of 2 or 3 to evaluate the effects on your structural stresses. And then carefully monitor the structural settlements or movements during construction to check the validity of your assumptions.