Specify analytic exponential and pseudoinverse of linear part of a SplitODEProblem

I have a system of ODE of the form

du/dt = A(t)*u + f(u)

The linear part corresponds to fast dynamics, in comparison to the slow dynamics of the non-linear part.

A(t) is actually so simple, that I can provide an analytic expression for the time-dependent exponential as well as for pseudoinverse.
Is there a possibility to specify the analytical expressions for the exponential and pseudoinverse with SplitODEProblems?

Are there Exponential Integrators that can handle time-dependent linear part?
(The change of the linear part is actually the slowest time-scale in my case.)

This is not currently a way to provide this.