If all numbers are real, it is decidable. This follows, for example from the general result in the paper

Vorobʹev, N. N., Jr. Deciding the consistency of a system of inequalities...

https://link.springer.com/chapter/10.1007%2F978-1-4612-0441-1_33.

If complex numbers are allowed it is not clear to me what the answer is.
Chebotarev has a generalization of Sturm's theorem to functions
of the form $P(x,\cos x,\sin x)$ where $P$ is a polynomial, but I do not
know a reference for a general result with complex $\lambda_k$. I suspect it might be wrong: with complex $\lambda$ the question might be undecidable.