...is sufficiently big. So take your hopefuly definition $\phi$, and we may as well make the well-ordered bit some predicate $(P, <)$. If $\phi$ has models were $(P, <)$ has size (cofinal in) the Hanf number, build a blueprint $\phi$ so the indiscernibles are in $P$. This will contradict the well-orderedness by, e.g., plugging in an I with lots of automorphisms.

@JoelDavidHamkins Perhaps another way to see this through Hanf numbers is by building indiscernibles. Morley's Omitting Types Theorem (which I recently learned is due to Chang in the uncountable case) says that if $\phi \in L_{\kappa, \omega}$ has models of size (cofinal in) $\beth_{(2^\kappa)^+}$, then it has arbitrarily large models. The proof goes by building a blueprint $\Phi$ that builds model $EM_\tau(T, \Phi)$ of $\phi$ with $I$ as indiscernibles. You can modify this to build the indiscernibles in whatever definable set you like (or more) as long as the definable set...

I don't have access to those articles, but is the issue that representing a complex number in the algebraic closure as 'a+bi' with a, b real requires choice?

It seems like the definability is critical here; a soon-to-be-completed (hopefully) preprint with Dimopoulos, GItman, and Magidor shows (among other things) that, over GBC, the two are equivalent: "Ord is subtle" and "Every logic has a stationary class of weakly compact cardinals". Also, global choice is necessary here, with our specific example being $\mathbb{L}^2$

It feels like the test property at $\kappa$ corresponds to the Tarski-Vaught Test for the cardinality quantifiers at $\kappa$ and below. So my guess is no, and that a counter-example can be cooked out of some logic outside of $\bL(Q_\kappa)$ and inside $\bL^2$ like cofinality quantifiers or $\bL(aa)$. But this elementarity for $\bL^2$ is hard for me to wrap my head around enough to cook up a proof...

Just to be clear, what is your notion of second-order substructure? Is it the one where you just look at formulas with element free-variables (and no set free-variables)?

One more addendum: I think averageability of some honest/unavoidable use of a universal quantifier for all structures is going to be very hard. As M gets big, the different between $P(\prod M)$ and $\prod P(M)$ is going to get bigger and bigger, so you'll keep missing out on more subsets that you need to universally quantify over. This is why I think an example would be very instructive, since you'll start to see where averageability fails.

An addendum to the $\Delta_1$ observation above: a better way of looking at it is that existential second-order sentences are preserved by ultraproducts is that the are PC-classes and reduct commutes with ultraproducts.