I have been thinking about this question since it was bumped up. I am burned out now, so I thought I would post a few basic things that I noted. 

First, as you noted that any function $f \in \mathcal F$ would reach a constant value after some point. Also that value will obviously be the maximum value of the function. So suppose that $f:\omega_1 \rightarrow \omega_1$ maximises at a value $v_f$. Now it seems to me that this variable $v_f$ may be of some significance for the given question. 

For example, let $\mathcal F_\alpha \subset \mathcal F$ ($\alpha<\omega_1$) denote the collection of functions $f \in \mathcal F$, which satisfy the additional property the $v_f < \alpha$. Your original question was **(A)** Giving a function $\psi : \mathcal F \to \mathcal K$ (satisfying certain properties of course). Now I think we can also consider a sub-question: **(B)** Can one give a function $\psi_\alpha : \mathcal F_\alpha \to \mathcal K$ for any arbitrary $\alpha < \omega_1$ ($\psi_\alpha$ and $\mathcal K$ satisfying properties as described in question).

For example, here is an example of a function $\psi_{\omega^2} : \mathcal F_{\omega^2} \to \mathcal K$. I think, I can give a similar description for $\psi_{\omega^3} : \mathcal F_{\omega^3} \to \mathcal K$. I have not added it for the sake of brevity, but if you think the example for $\psi_{\omega^2}$ is not illustrative enough, then I will add it.  

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An example for $\psi_{\omega^2}$. Given a function $f \in \mathcal F_{\omega^2}$, first determine the maximum value $v_f$ of the function. Now suppose that the maximum value $v_f$ is of the form $\omega \cdot a+b$ (with $a \in \mathbb{N}^+$, $b \in \mathbb{N}$). Denote $\psi_{\omega^2} (f)$ as $F$. Now define $F$ as follows:
$$F(x)=\omega \cdot a+b=v_f \qquad \mathrm{for} \quad  x > \omega \cdot(a+1)$$
$$F(x)=\omega \cdot a \qquad \mathrm{for} \quad \omega\cdot a < x \leq \omega \cdot (a +1)$$
$$F(x)=\omega \cdot n \qquad \mathrm{for} \quad \omega\cdot n < x \leq \omega \cdot (n+1)$$
$$F(x)=f(x) \qquad \mathrm{for} \quad x \leq \omega $$
In the third line, we have $1 \leq n < a$. (**EDIT :** Made a correction in the first and second equation) 

One line of thinking is to see whether we can keep giving the function $\psi_\alpha$ or not. If no, then what is the point at which we can no longer do that (the answer to **(A)**, the question in OP, would be negative in that case). If yes, in that case the answer to **(B)** would be positive. Also, in that case, what would be the general requirements/obstacles at each level (to determine whether **(B)** implies **(A)** or not). Anyway, just a suggestion that might perhaps be useful (and, to be fair, possibly not useful).