In the following paper
Howard, Paul E., Limitations on the Fraenkel-Mostowski method of independence proofs, J. Symb. Log. 38, 416-422 (1973). ZBL0325.02043.
The author alludes to a construction by David Pincus that shows the independance of the axiom of choice for sets os finite sets from the axiom of choice for sets of well ordered sets. Can anybody point me towards a paper by Pincus that contains such a construction?
In the Howard–Rubin bible, they mention the model $\cal M1(\langle\omega_1\rangle)$ on page 147, as a model where Form 85 (choice from families of countable sets) fails, but Form 62 (choice from families of finite sets) holds as a consequence of linear orderings. Unfortunately, there's no clear reference to the construction. So here is an outline.
The idea of the construction is to add $\omega\times\omega_1$ Cohen reals, then take permutations that only move each $\omega\times\{\alpha\}$ within itself. So we are only allowed to permute each $\omega$-block separately. And take things preserved when fixing pointwise countably many blocks.
Easily, in that case, the sequence of $\omega$-blocks is symmetric, but the standard argument also shows you cannot choose from them. Therefore choice from families of countable sets fails, so choice from families of well-ordered sets fails as well.
There is a rough outline and references to the proof that every set can be linearly ordered in this model in the book. So I leave that part to you.
(As an easy exercise, the fact the forcing we use is ccc, and the filter of groups is $\sigma$-closed, we immediately get Dependent Choice in the outcome model!)