How to determine whether a proof requires "higher-order reasoning techniques"?
Briefly, every theorem stated in first-order logic has a first-order proof.

In his book "An Introduction to Mathematical Logic and Type Theory", Peter B. Andrews develops both first-order logic and a system of higher-order logic Q0, which is generally considered to be the theory basis of modern higher-order provers. (See the introduction to the HOL logic for example.)

For Q0 and similar systems, Andrews shows that the higher-order logics he describes can be considered as conservative extensions of first-order logic and writes (second edition, p. 259) that, "In summary, every first-order theorem of type theory has a first-order proof."

Given your practical concerns though, I also quote the following paragraph:

"However, some theorems of first-order logic can be proved most efficiently by using concepts which can be expressed only in higher-order logic. Examples may be found in [Andrews and Bishop, 1996] and [Boolos, 1998, Chapter 25]. Statman proved [Statman, 1978, Proposition 6.3.5] that the minimal length of a proof in first-order logic of a wff of first-order logic may be extraordinarily longer than that the minimal length of a proof of the same wff in second-order logic. A related result by Godel [Godel, 1936] is that in general 'passing to the logic of the next higher order has the effect, not only of making provable certain propositions that were not provable before, but also of making it possible to shorten, by an extraordinary amount, infinitely many of the proofs already available'. A complete proof of this may be found in [Buss, 1994]."