I've combined stabilized and non stabilized materials in Wa handles a bunch of times and never had any problems from inconsistent movement/shrinking. I won't say it's a negligible detail - the type of unstabilized wood you are using , how dry the wood is, and how humid an environment you're in could be factors that make a difference. That said, though, if you're careful about your techniques and how you join the two materials, you should be fine unless your pieces are especially thin. I'd recommend using a good epoxy that has some light flex at the joint to help keep things in check if there is any subtle movement. (I personally use G-Flex)
I'm by no means an expert on stabilized woods, but couple other points others have shared with me over the years --- If you're working from a block of stabilized wood, I've been told it's generally a good idea to trim the outer 1/8 inch off and consider that scrap. The outer edges being most prone to warping as a byproduct of the chemical reactions and forces put on the material in the stabilizing process (vacuum, heat etc). Precut scale blanks, obviously that advice is irrelevant.
The other general tip that applies to any thin woodworking project (Stabilized or not) is to try to treat both sides of your material as consistently as possible. As an unrelated illustrative example -- when working with veneer....when you glue it down to a substrate, it wants to curl and the power of that warping force is surprisingly strong. By wetting the visible face with water or softener, it helps prevent the veneer from cupping or pulling on the other material. Sort of the equal and opposite forces rules of physics. Where possible, the same principle is usually good to follow and helps prevent problems here too.
Other info, though I'd guess overkill:
flatsawn wood will move far more than quartersawn.
You can find tables that show formulas to calculate wood movement based on species. Dense hardwoods properly dried are going to be more stable in general.....
In dry woods, most movement is going to be across grain - side to side - rather than end to end. Because of that, the stress on your glue joint between materials is lateral. That kind of movement is unlikely to shear your joint, and since you will have glued your blanks to your tang this being a Western handle, you'll have a lot of surface area preventing that kind of lateral movement in your woods. More reason not to worry.
an oil finish that provides a moisture barrier will help prevent problems down the road too