Structurally this will only work if the first argument of FixedPoint gives output in the same form the second argument (the start value) has - to allow for iteration. Can you show please how Apply[fun] produces a list of length 6? F1 and F0 are {6,4} matrices, to evaluate them in a scalar function like Exp one needs 2 vectors to absorb them into a scalar

In[99]:= {1, 2, 3, 4, 5, 6}.F0.{7, 8, 9, 10, 11, 12}
Out[99]= 205.469

In[100]:= {1, 2, 3, 4, 5, 6}.F1.{7, 8, 9, 10, 11, 12}
Out[100]= 184.469

but the above definitions apply only one Dot-Operator. Then one needs to give instantiations of the scalar values z1 and y to get it started, if the algebra has been fixed.