Just to mark this question as answered: The answer is yes. Some details follow.
The basic idea was generously provided by the anonymous referee of a short note (joint work with Paolo Leonetti) that has been recently accepted for publication in some journal (*). The key ingredient is Theorem 3 from:
J.-H. Evertse, The number of solutions of decomposable form equations, Invent. Math. 122 (1995), No. 3, 559–601,
which yields, for a fixed finite set of primes $\mathcal S$, an effective bound on the number of non-degenerate solutions of an $\mathcal S$-unit equation in $k$ variables (over the additive group of the rationals).
(*) I'm still trying to understand how to avoid self-promotion in situations like this...