That's a comment but it got a bit too long.
As far as I know, classically, the approximation property deals with bounded linear operators.
But let us assume your point for a minute.
If it is not "bounded linear" then what is you definition of "bounded" operator? E.g., 1) a bounded function, that is a function with bounded range. But a bounded linear operator is not a bounded function unless it is zero, so it'd be inconsistent with the standard definition of AP; or maybe 2) a function, sending bounded sets to bounded sets.
If 2), you may want to approximate it by linear finite-rank operators. But even in the 1-dimensional case you cannot do it (e.g., try to approximate $x^2$). Alternatively, you may want to approximate it by (non-linear) functions with finite-dimensional range, which is, of course, of no interest in a finite-dimensional space.
However, it is true that people consider some versions of the AP for non-linear operators. See, e.g., Question 4 in this article by Godefroy and Ozawa: