(1) A choice from a set $X\subseteq {\cal P}(Y)$ is a function $a:X\rightarrow\bigcup X$ with $a(x)\in x$ for all $x\in X$.
(2) A tree is a function $Tf:T\rightarrow {\cal P}(T)$ giving a (possibly infinite) set of children of argument vertex. (Various definitions of tree should work, as long as no branches are longer than $\omega$. E.g., a well-founded poset $(T,<)$, where for every $x\in T: \{y\in T\mid y< x\}$ is finite and totally ordered by $<$. A branch is any maximal $<$-chain. $Tf(x)$ is then the set of minimal elements in $\{y\in T\mid y>x\}$.) $Tf$ extends pointwise to sets: $Tf(X) = \bigcup_{v\in X}Tf(v)$.
(3) Now, all vertices of our specific $T$ are mutually disjoint, finite sets. With each $v\in T$, there is associated a function $f_v: \prod{Tf(v)} \rightarrow v$, such that for each subtree $F$ of $T$ with only finitely many infinite branches, each choice $\alpha$ from $Tf(F) \setminus F$ can be extended to a choice $\beta\supseteq\alpha$ from $Tf(F)\cup F$ such that for all $v\in F$:
(*) $\beta(v) = f_{v}(\prod_{s\in Tf(v)}\beta(s)).$
(4) Is it true that then
there is also a choice $\beta$ from the whole $T$ satisfying (*) for all $v\in T$? In summary:
Given the dependence (expressed by $f_v$) of the choice from each v on the choices made from all $Tf(v)$, is it possible to make a choice from all $v\in T$ respecting this dependence, provided that such a choice is possible from every finite collection $F$ of infinite branches of $T$?
(5) I have tried the following: For each finite collection (of infinite branches) $F$, let $C_F$ denote the set of choices (relatively to all choices from $Tf(F)\setminus F$) satisfying $(*)$ for all $v\in F$. For each $v\in T$, let $C_v$ denote the choices satisfying $(*)$ at $v$. The collection of all $C_F$s and $C_v$s has FIP, so let $U$ be an ultrafilter containing it. For $v_i\in v\in T$, let $v_i^*$ denote the choice of $v_i$ from $v$ combined with all choices from all other vertices. Since each $v\in T$ is finite, for every $v$ there is a unique $v_i\in v$ such that $v_i^*\in U$. However, I do not see an argument showing that the combination of all these $v_i$s would yield a choice satisfying (*) for all $v\in T$.
(6) If (4) is not true in general, are there any simple restrictions on $T$ or $f_v$ which make it true? (Requiring finite $Tf(v)$ suffices, but is not satisfactory for my purposes.)