Observations on cross grain construction:
I have made cross grain construction items for 40 years with very few problems. The most extreme example are many bread boards (for making bread) with breadboard ends. Most of these are about 18" wide, with quartersawn hard maple in 1-3/4" wide strips. Most had 3/16" wide Walnut strips between the Maple. This board is surfaced to about 7/16" and then tenoned on each end (about 3/16" thick by 5/8" long) and a Walnut end cap is fitted, about 1" x 1", plowed for the tenon.
These have been made with resorcinol, TB3, TB1 and TB2 as well as epoxy. They get a mineral oil finish, and little maintenance over the years. That means these are very prone to environmental changes in humidity, therefore wood movement.
I have had no joint failures, or even movement, period. 18" wide. I attribute this to the Q/S Maple and the surface area of the tenon. As well as the fact that these are almost never soaked in a sink or get more water than the occasional rinsing and drying.
So....if you were to increase the surface area of the joint on your two parts - splines, tenons, even rabbets - this should reduce or eliminate the movement, or the appearance of movement. A step between the two parts will also disguise any movement.
If absolute flush is required, then man-made core and veneers should be used.