What pure mathematics foundations should an applied mathematician have?

Linear algebra, for sure. Real analysis and "advanced calculus" also for sure. These concepts form the basis for numerical analysis, which is critically important for many applied fields. Even if a mathematician isn't directly working on numerical problems, there's typically a desire to develop concepts in a way that they are numerically solvable.

Probability is another field. Uncertainty quantification is hugely important in applied fields, and the interplay between probability, statistics, linear algebra, and dynamical systems cannot be overstated.

Lately, I am of the personal opinion that general algebra and things like representation and category theory are underutilized in applied fields. I have no formal basis for this, aside from my everyday work of research in engineering fields. A great many (i.e. almost all) papers completely ignore abstract formulations of the problems at hand, and they go through extreme procedural distortions to realize only moderate gains in solvability, accuracy, or computability... and so often the example still fail to represent real-world problems of interest in any meaningful way.