the reason why I suggest an applied math minor to OP is because OP mentioned financial engineering as a choice only.
you don't need it for a developer, programmer and etc role.
actually if you really take a minor or not is not a key point, but you have to have a series of basic course in order to be admitted into any solid MFE, and this list of course sum up to the amount similar to a minor in math + a minor in stat only. You may skip a couple of them, but skipping too much will close your door to most solid MFE (there are still chances for 2nd tier MFE or those more business one)
P.S. Multivariate Calculus, Linear Algebra, Calculus-base Probability, Proof-base Statistics, Regression, Time Series, Stochastic Process, Differential Equation, Numerical Analysis, Optimization, and etc.....