Does this lead to a contradiction within ZF?

If I understand you correctly, you want to say that if there is a surjection from $y$ onto $x$, then if there is any injection, there is one splitting the surjection.

This cannot lead to contradiction, since it follows from the Axiom of Choice. Another question would be whether or not this implies the Axiom of Choice. This is a bit more complicated.

Let "The Partition Principle" denote the statement "if there is a surjection from $y$ to $x$, then there is an injection". And what we're asking is: Does the Partition Principle implies AC provable from $sf ZF$? And the answer is that we don't know.