Here's my idea. We're going to construct a relation that you can apply Axiom of Choice to for obtaining the function you desire.

Pick some arbitrary

Construct a set of ordered pairs R s.t. it has two properties.

(1)

(2) if for some and <x^{+}, y'> s.t. y P_{r} y'

You know that there will always exist greater y's because P has no maximal unit.

Then from axiom of choice you can create a function from this relation and I believe it should have all the properties you want.