It seems unlikely that such a bound can hold in general. For instance, it fails for round spheres, which have very large multiplicity of eigenvalues. In fact, the eigenvalue counting function has jumps of order $\lambda^{n-1}$. On a round 2-sphere, for a constant eigenvalue $\lambda $, the second term in your estimate will vary on order $\lambda$, which makes a uniform bound on the left hand side impossible.

For generic metrics where closed geodesics are "sparse," it's possible to strengthen the standard Weyl law but I don't think there is any hope of getting an estimate as strong as you want. I believe the state of the art refinement of Weyl's law is due to Canzani and Galkowski, so that might be a good reference.

It seems unlikely that such a bound holds except in very special cases. For instance, it fails for round spheres, which have very large multiplicity of eigenvalues. In fact, for spheres the eigenvalue counting function has jumps of order $\lambda^{n-1}$. On a round 2-sphere, for a constant eigenvalue $\lambda $, the second term in your estimate will vary on order $\lambda$, which makes a uniform bound on the left hand side impossible.

For generic metrics where closed geodesics are "sparse," it's possible to strengthen the standard Weyl law, but the refinement isn't strong enough to give uniform bounds on the difference between eigenvalues and the asymptotic formula from Weyl's law. I believe the state of the art refinement of Weyl's law is due to Canzani and Galkowski, and their paper is a good reference. The relevant result is Theorem 7 on page 12.