This is usually called the magnus expansion method and has a nice literature in numerical analysis. Kato also used this method to show the existence of solutions in the hyperbolic case.
I would say that strong resolvent continuity and a sufficiently big common domain is sufficient in your case. See Section 5.3 in Pazy.
I can also give a related self-reference, where also the investigation of this product appears and some of the ideas are explained in a simpler situation.